On the Fractional NLS Equation and the Effects of the Potential Well’s Topology

Silvia Cingolani; Marco Gallo

doi:10.1515/ans-2020-2114

Publicly Available Published by De Gruyter November 12, 2020

On the Fractional NLS Equation and the Effects of the Potential Well’s Topology

Silvia Cingolani and Marco Gallo

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2020-2114

Abstract

In this paper we consider the fractional nonlinear Schrödinger equation

ε 2 ⁢ s ⁢ ( - Δ ) s ⁢ v + V ⁢ ( x ) ⁢ v = f ⁢ ( v ) , x ∈ ℝ N ,

where s∈(0,1), N≥2, f is a nonlinearity satisfying Berestycki–Lions type conditions and V∈C⁢(ℝN,ℝ) is a positive potential. For ε>0 small, we prove the existence of at least cupl⁢(K)+1 positive solutions, where K is a set of local minima in a bounded potential well and cupl⁢(K) denotes the cup-length of K. By means of a variational approach, we analyze the topological difference between two levels of an indefinite functional in a neighborhood of expected solutions. Since the nonlocality comes in the decomposition of the space directly, we introduce a new fractional center of mass, via a suitable seminorm. Some other delicate aspects arise strictly related to the presence of the nonlocal operator. By using regularity results based on fractional De Giorgi classes, we show that the found solutions decay polynomially and concentrate around some point of K for ε small.

Keywords: Fractional Laplacian; Nonlinear Schrödinger Equation; Concentration Phenomena, Pohozaev Identity; Relative Cup-Length

MSC 2010: 35A15; 35B25; 35J20; 35Q55; 35R11; 47J30; 58E05

1 Introduction

In [39] Laskin developed a new extension of the fractality concept in quantum physics and formulated the fractional nonlinear Schrödinger (fNLS for short) equation

(1.1) i ⁢ ℏ ⁢ ∂ t ⁡ ψ = ℏ 2 ⁢ s ⁢ ( - Δ ) s ⁢ ψ + V ⁢ ( x ) ⁢ ψ - f ⁢ ( ψ ) , ( t , x ) ∈ ( 0 , + ∞ ) × ℝ N ,

where s∈(0,1), N>2⁢s, the symbol (-Δ)s⁢ψ=ℱ-1⁢(|ξ|2⁢s⁢ℱ⁢(ψ)) denotes the fractional power of the Laplace operator defined via Fourier transform ℱ on the spatial variable, ℏ designates the usual Planck constant, V is a real potential and f is a Gauge invariant nonlinearity, i.e., f⁢(ei⁢θ⁢ρ)=ei⁢θ⁢f⁢(ρ) for any ρ,θ∈ℝ. The nonlocal equation (1.1) was derived as extension of the classical NLS equation (s=1), replacing the path integral over Brownian motions (random motions seen in swirling gas molecules) to Lévy flights (a mix of long trajectories and short, random walk models found in turbulent fluids). The wave function ψ⁢(x,t) represents the quantum mechanical probability amplitude for a given unit mass particle to have position x and time t, under the confinement due to the potential V, and |ψ|2 is the corresponding probability density. We refer to [39, 40] for a detailed discussion of the physical motivation of the fNLS equation.

Special solutions of the equation (1.1) are given by the standing waves, i.e., factorized functions

ψ ⁢ ( t , x ) = e i ⁢ E ⁢ t ℏ ⁢ v ⁢ ( x ) , E ∈ ℝ .

For small ℏ>0, these standing waves are usually referred to as semiclassical states and the transition from quantum physics to classical physics is somehow described letting ℏ→0 (see [45]).

Without loss of generality, shifting E to 0 and denoting ℏ≡ε, the search of semiclassical states leads to investigate the following nonlocal equation:

(1.2) ε 2 ⁢ s ⁢ ( - Δ ) s ⁢ v + V ⁢ ( x ) ⁢ v = f ⁢ ( v ) , x ∈ ℝ N ,

when V is positive and ε>0 is small.

In the limiting case s=1 the semiclassical analysis of NLS equations has been largely investigated, starting from the seminal paper [35]. By means of a finite dimensional reduction, Floer and Weinstein proved the existence of positive spike solutions to the 3D cubic NLS equation, concentrating at each nondegenerate critical point of the potential V (see also [42]). Successively, refined variational techniques were implemented to study singularly perturbed elliptic problems in entire space: several existence results of positive spike solutions to the NLS equation in a semiclassical regime are derived under different assumptions on the potential and the nonlinear terms. We confine to mention [43, 49, 27, 3, 28, 14, 15, 17, 18] and references therein. In [22, 4, 21] topological invariants were used to derive multiplicity results, in the spirit of well-known results of Bahri and Coron [6] and Benci and Cerami [11] for semilinear elliptic problems with Dirichlet boundary condition. Precisely, in [22] it has been proved that the number of positive solutions of the stationary NLS equation is influenced by the topological richness of the set of global minima of V. Some years later, using a perturbative approach, Ambrosetti, Malchiodi and Secchi [4] obtained a multiplicity results for NLS equation with power nonlinearity, assuming that the set of critical points of V is nondegenerate in the sense of Bott.

More recently, in [21] Jeanjean, Tanaka and the first author improved the result in [22], relating the number of semiclassical standing waves solutions to the cup-length of K, where K is a set of local (possibly degenerate) minima of the potential, under almost optimal assumptions on the nonlinearity (see also the recent paper [24] in the context of nonlinear Choquard equations).

When s∈(0,1), the search of semiclassical standing waves for the fNLS equation has been firstly considered by Dávila, Del Pino and Wei in [26] under the assumptions f⁢(t)=|t|p-2⁢t, with 2<p<2s*, where 2s*:=2⁢NN-2⁢s is the Sobolev critical exponent, and V∈C1,α⁢(ℝN) is bounded. Using Lyapunov–Schmidt reduction inspired by [35, 42], they showed the existence of a positive spike solution whose maximum point concentrates at some nondegenerate critical point of V; this approach relies on the nondegeneracy property of the linearization at the positive ground state shown by Frank, Lenzmann and Silvestre [37]. Successively, inspired by [27, 14], variational techniques were employed to derive existence of spike solutions concentrating at local minima of V, see [46, 2, 5] and references therein (see also [44] where global assumptions on V are considered). We cite also [32] in which necessary conditions are given to the existence of concentrating solutions.

A first multiplicity result for the (fNLS) equation is obtained in [34], inspired by [22]. Precisely, letting K be the set of global minima of V, Figueiredo and Siciliano proved that the number of positive solutions of equation (1.2), when f satisfies monotonicity and Ambrosetti–Rabinowitz condition, is at least given by the Ljusternik–Schnirelmann category of K: here the search of solutions of (1.2) can be reduced to the study of the (global) level sets of the Nehari manifold, where the energy functional is restricted, and to deformation arguments valid on Hilbert manifolds without boundary. See also [1] where the Ambrosetti–Rabinowitz condition is dropped. In [20], moreover, Chen implemented a Lyapunov–Schmidt reduction for nondegenerate critical points of V and power-type functions f in order to get multiplicity results related to the cup-length, extending the results of [4].

In the present paper we are interested to prove multiplicity of positive solutions for the fNLS equation (1.2) when ε is small, without monotonicity and Ambrosetti–Rabinowitz conditions on f, nor nondegeneracy and global conditions on V.

On the potential V we assume

V ∈ C ⁢ ( ℝ N , ℝ ) ∩ L ∞ ⁢ ( ℝ N ) , V¯:=infℝN⁡V>0 (see also Remark 1.3),
there exists a bounded domain Ω such that

m 0 := inf Ω ⁡ V < inf ∂ ⁡ Ω ⁡ V ,

by the strict inequality and the continuity of V, we can assume that ∂⁡Ω is regular. We define K the set of local minima

(1.3) K := { x ∈ Ω ∣ V ⁢ ( x ) = m 0 } .

On f we assume

Berestycki–Lions assumptions ([12]) with respect to m0, that is,
1. f ∈ C ⁢ ( ℝ , ℝ ) ,
2. lim t → 0 ⁡ f ⁢ ( t ) t = 0 ,
3. lim t → + ∞ ⁡ f ⁢ ( t ) | t | p = 0 for some p∈(1,2s*-1), where we recall 2s*=2⁢NN-2⁢s,
4. F ⁢ ( t 0 ) > 1 2 ⁢ m 0 ⁢ t 0 2 for some t0>0, where F⁢(t):=∫0tf⁢(s)⁢𝑑s,
f ⁢ ( ξ ) = 0 for ξ≤0.

On f we further assume

f ∈ C loc 0 , γ ⁢ ( ℝ ) for some γ∈(1-2⁢s,1) if s∈(0,1/2].

We remark that (f3) is needed only to get a Pohozaev identity (see [16, Proposition 1.1]).

Setting u:=v(ε⋅), (1.2) can be rewritten as

(1.4) ( - Δ ) s ⁢ u + V ⁢ ( ε ⁢ x ) ⁢ u = f ⁢ ( u ) , x ∈ ℝ N ,

thus the equation

(1.5) ( - Δ ) s ⁢ U + a ⁢ U = f ⁢ ( U ) , x ∈ ℝ N ,

for some a>0, becomes a formal limiting equation, as ε→0, for (1.4).

It is standard that weak solutions to (1.4) correspond to critical points of the C1-energy functional

I ε ⁢ ( u ) := 1 2 ⁢ ∫ ℝ N | ( - Δ ) s 2 ⁢ u | 2 ⁢ 𝑑 x + 1 2 ⁢ ∫ ℝ N V ⁢ ( ε ⁢ x ) ⁢ u 2 ⁢ 𝑑 x - ∫ ℝ N F ⁢ ( u ) ⁢ 𝑑 x , u ∈ H s ⁢ ( ℝ N ) ,

where Hs⁢(ℝN) is the fractional Sobolev space. We remark that, because of the general assumptions on f, we cannot take advantage of the boundedness of the functional from above and below, nor of Nehari-type constraint. Therefore in the present paper we combine reduction methods and penalization arguments in a nonlocal setting: in particular, as in [21, 24], the analysis of the topological changes between two level sets of the indefinite energy functional Iε in a small neighborhood 𝒳ε,δ of expected solutions is essential in our approach. With the aid of ε-independent pseudo-differential estimates, we detect such a neighborhood, which will be positively invariant under a pseudo-gradient flow, and we develop our deformation argument in the context of nonlocal operators. To this end, we introduce two maps Φε and Ψε between topological pairs: we emphasize that to define such maps, a center of mass Υ and a functional Pa which is inspired by the Pohozaev identity are crucial.

With respect to the local case, several difficulties arise linked to special features of the nonlocal nature of the problem: among them we have the polynomial decay of the least energy solutions of the limiting problems, the weak regularizing effect of the fractional Laplacian, the lack of general comparison arguments, the differences between the supports of a function and of its Fourier transform, and the lack of the standard Leibniz formula (see e.g. [44, 13]).

We highlight that, for fractional equations, the nonlocal part strongly influences the decomposition of the space and this makes quite delicate to use truncating test functions and perform the localization of the centers of mass. In the present paper we introduce a new fractional local center of mass by means of suitable seminorm, stronger than the usual Gagliardo seminorm in a bounded set and we need to implement new ideas to overcome the above obstructions.

Our main result is the following theorem.

Theorem 1.1.

Suppose N≥2 and that (1)–(2), (f1)–(f3) hold. Let K be defined by (1.3). Then, for sufficiently small ε>0, equation (1.2) has at least cupl⁢(K)+1 positive solutions, which belong to C0,σ⁢(RN)∩L∞⁢(RN) for some σ∈(0,1).

Here cupl⁢(K) denotes the cup-length of K defined by the Alexander–Spanier cohomology with coefficients in some field 𝔽 (see Section 5).

Remark 1.2.

Notice that the cup-length of a set K is strictly related to the category of K. Indeed, if K=SN-1 is the (N-1)-dimensional sphere in ℝN, then cupl⁢(K)+1=cat(K)=2; if K=TN is the N-dimensional torus, then cupl⁢(K)+1=cat(K)=N+1. However, in general, cupl⁢(K)+1≤cat(K).

A multiplicity result in the limiting case s=1, similar to Theorem 1.1, is contained in [21].

Remark 1.3.

Observe that, arguing as in [14, 15, 17], we could omit the assumption that V is bounded from above in Theorem 1.1. For the sake of simplicity, we assume here the boundedness of V.

By using recent regularity results based on fractional De Giorgi classes and tail functions (see [25]), we are able to prove also the following theorem.

Theorem 1.4.

Let (εn)n∈N with εn→0+ as n→+∞. For sufficiently large n∈N, in the assumptions of Theorem 1.1, let vεn be one of the cupl⁢(K)+1 solutions of equation (1.2). Then, up to a subsequence, (vεn)n∈Nconcentrates in K as n→+∞. More precisely, for each n∈N there exists a maximum point xεn of vεn such that

lim n → + ∞ ⁡ d ⁢ ( x ε n , K ) = 0 .

In addition, vεn(εn⋅+xεn) converges in Hs⁢(RN) and uniformly on compact sets to a least energy solution of

(1.6) ( - Δ ) s ⁢ U + m 0 ⁢ U = f ⁢ ( U ) , U > 0 , U ∈ H s ⁢ ( ℝ N ) ,

and, for some positive C′,C′′ independent on n∈N, we have the uniform polynomial decay

C ′ 1 + | x - x ε n ε n | N + 2 ⁢ s ≤ v ε n ⁢ ( x ) ≤ C ′′ 1 + | x - x ε n ε n | N + 2 ⁢ s for x ∈ ℝ N .

The paper is organized as follows. In Section 2 we recall some basic notions on the nonlocal framework. In Section 3 we overview some results on the limiting equation (1.5) and we introduce a new fractional center of mass Υ, by means of a suitable seminorm; we postpone to Appendix A the proof of the uniform polynomial decay of the solutions of (1.5), since it shares some argument with the proof of Theorem 1.4. Section 4 is the main core of the article, where we introduce a penalized functional and prove a deformation lemma on a neighborhood of expected solutions; moreover, we build suitable maps Φε, Ψε essential in the proof of the multiplicity of solutions. In Section 5 we prove Theorem 1.1 by the use of the deformation lemma and the built maps applied to the theory of relative category and relative cup-length, of which we briefly recall the definitions. Finally, we prove Theorem 1.4 by using regularity results based on fractional De Giorgi classes.

2 Preliminaries

Let N>2⁢s. Set the Gagliardo seminorm

[ u ] ℝ N 2 := ∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

we can define the fractional Sobolev space ([31, Section 2])

H s ⁢ ( ℝ N ) := { u ∈ L 2 ⁢ ( ℝ N ) ∣ [ u ] ℝ N < + ∞ }

and set for every u∈Hs⁢(ℝN),

∥ u ∥ H s ⁢ ( ℝ N ) 2 := ∥ u ∥ L 2 ⁢ ( ℝ N ) 2 + [ u ] ℝ N 2 ;

here ∥⋅∥Lq⁢(Ω) denotes the Lq-Lebesgue norm for q∈[1,+∞], and we will often write ∥⋅∥q:=∥⋅∥Lq⁢(ℝN). Moreover, we define the fractional Laplacian

( - Δ ) s ⁢ u ⁢ ( x ) := C ⁢ ∫ ℝ N u ⁢ ( x ) - u ⁢ ( y ) | x - y | N + 2 ⁢ s ⁢ 𝑑 x ,

where C (as the following appearing constants) depends on N and s, and the integral is in the principal value sense; the following equality holds ([31, Proposition 3.6]):

(2.1) ∥ ( - Δ ) s 2 ⁢ u ∥ 2 2 = C ′ ⁢ [ u ] ℝ N 2

and more generally (by polarization identity)

∫ ℝ N ( - Δ ) s 2 ⁢ u ⁢ ( - Δ ) s 2 ⁢ v ⁢ 𝑑 x = C ′ ⁢ ∫ ℝ N ∫ ℝ N ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ ( v ⁢ ( x ) - v ⁢ ( y ) ) | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y .

We further have (see [31, Theorem 6.5])

H s ⁢ ( ℝ N ) ↪ L q ⁢ ( ℝ N )

for q∈[2,2s*] and, for u∈Hs⁢(ℝN),

(2.2) ∥ u ∥ L 2 s * ⁢ ( ℝ N ) ≤ C ′′ ⁢ ∥ ( - Δ ) s 2 ⁢ u ∥ L 2 ⁢ ( ℝ N ) .

Moreover, if q∈[2,2s*), we have ([31, Corollary 7.2])

H s ( ℝ N ) ↪ ↪ L loc q ( ℝ N )

in the sense that for every (un)n bounded in Hs⁢(ℝN), and for every A⊂ℝN bounded and regular enough (∂⁡A Lipschitz), we have that (un)n restricted to A admits a convergent subsequence in Lq⁢(A).

It will be useful to work also with

∥ u ∥ H ε s ⁢ ( ℝ N ) 2 := ∥ ( - Δ ) s 2 ⁢ u ∥ 2 2 + ∫ ℝ N V ⁢ ( ε ⁢ x ) ⁢ u 2 ⁢ 𝑑 x

which is an equivalent norm on Hs⁢(ℝN), thanks to the positivity and the boundedness of V and (2.1); the space Hεs⁢(ℝN) is defined straightforwardly.

Finally, we will make use of the following mixed Gagliardo seminorm

[ u ] A 1 , A 2 2 := ∫ A 1 ∫ A 2 | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y , [ u ] A := [ u ] A , A ,

for any A1,A2,A⊂ℝN and u∈Hs⁢(ℝN); by using that the function

φ u ⁢ ( x , y ) := | u ⁢ ( x ) - u ⁢ ( y ) | | x - y | N 2 + s , x , y ∈ ℝ N , x ≠ y ,

satisfies φu+v≤φu+φv and [u]A1,A2=∥φu∥L2⁢(A1×A2), we have that [u]A1,A2 is actually a seminorm.

For any u∈Hs⁢(ℝN) and A⊂ℝN it will be useful to work also with the following norms:

(2.3) ∥ u ∥ A 2 := ∥ u ∥ L 2 ⁢ ( A ) 2 + [ u ] A , ℝ N 2

and

(2.4) || | u | || A := ∥ u ∥ L p + 1 ⁢ ( A ) + ∥ u ∥ A ,

where p is introduced in assumption (f1.3). We highlight that ∥u∥ℝN=∥u∥Hs⁢(ℝN), but generally ∥u∥A≥∥u∥Hs⁢(A) for A≠ℝN (see [31, Section 2] for a definition of Hs⁢(A)). By Hs⁢(A)↪Lp+1⁢(A) ([31, Theorem 6.7]) the norms ∥⋅∥A and |||⋅|||A are equivalent: on the other hand, the constant such that |||u|||A≤CA⁢∥u∥A depends on A, thus not useful for ε-dependent sets A=A⁢(ε). This is why we will make direct use of |||⋅|||A.

Moreover, we denote by

[ u ] C 0 , σ ⁢ ( A ) := sup x , y ∈ A x ≠ y ⁡ | u ⁢ ( x ) - u ⁢ ( y ) | | x - y | σ

the usual seminorm in Hölder spaces for σ∈(0,1]. We allow σ>1 by simply writing Cσ⁢(ℝN).

Before ending this section, we highlight that the assumptions on the function f imply the following standard fact, which will be extensively used throughout the paper: for each q≥p and β>0 there exists a Cβ>0 such that

(2.5) | f ⁢ ( t ) | ≤ β ⁢ | t | + C β ⁢ | t | q and | F ⁢ ( t ) | ≤ C ⁢ ( β ⁢ | t | 2 + C β ⁢ | t | q + 1 ) .

Further Notation.

We highlight that, all throughout the paper, we will assume N≥2, and the constants C,C′ appearing in inequalities may change from a passage to another. To avoid cumbersome notations, we will not stress the dependence of such constants, which will be based only on the fixed quantities. Moreover, by ≈ we will mean approximately equal to.

We will write BR⁢(x0) for the ball, centered in x0∈ℝN, with radius R>0; in particular, BR:=BR⁢(0). Moreover,

A δ := { x ∈ X ∣ d ⁢ ( x , A ) ≤ δ }

for any A⊂(X,d) metric space. In addition, we will sometimes write ∁⁢(A):=ℝN∖A for any A⊂ℝN to avoid cumbersome notation.

Finally, for every A⊂B⊂ℝN, we will write

A ≺ ϕ ≺ B

to indicate a Urysohn-type regular function ϕ∈Cc∞⁢(ℝN) such that ϕ|A=1 and ϕ|ℝN∖B=0.

3 Limiting Equation

3.1 A Single Equation

Consider

(3.1) ( - Δ ) s ⁢ U + a ⁢ U = f ⁢ ( U ) , x ∈ ℝ N ,

with a>0. Weak solutions of (3.1) are known to be characterized as critical points of the C1-functional La:Hs⁢(ℝN)→ℝ

L a ⁢ ( U ) := 1 2 ⁢ ∥ ( - Δ ) s 2 ⁢ U ∥ 2 2 + a 2 ⁢ ∥ U ∥ 2 2 - ∫ ℝ N F ⁢ ( U ) ⁢ 𝑑 x , U ∈ H s ⁢ ( ℝ N ) .

Set moreover the Pohozaev functional Pa:Hs⁢(ℝN)∖{0}→ℝ+

P a ⁢ ( U ) := ( 2 ⁢ N N - 2 ⁢ s ⁢ ∫ ℝ N F ⁢ ( U ) ⁢ 𝑑 x - a 2 ⁢ ∥ U ∥ 2 2 ∥ ( - Δ ) s 2 ⁢ U ∥ 2 2 ) + 1 2 ⁢ s , U ∈ H s ⁢ ( ℝ N ) , U ≠ 0 .

By [16, Proposition 1.1] we have that the Pohozaev identity

P a ⁢ ( U ) = 1

holds for each nontrivial solution U of (3.1). To reach this claim we need the solutions to be regular enough, fact that is given by (f3). We observe that the functional Pa will be of key importance for estimating La from below, see Lemma 3.3 and Lemma 3.5.

In addition, we recall the following result by [16, Theorem 1.2].

Theorem 3.1 (Existence of a Pohozaev Minimum).

Assume (f1) with respect to a>0 and (f2). Let

C p ⁢ o , a := inf ⁡ { L a ⁢ ( U ) ∣ U ∈ H s ⁢ ( ℝ N ) ∖ { 0 } , P a ⁢ ( U ) = 1 } .

Then there exists a positive minimizer, which is a weak solution of (3.1); this minimizer is generally not unique, even up to translation. Moreover, Cp⁢o,a>0.

We further denote by

E a := inf ⁡ { L a ⁢ ( U ) ∣ U ∈ H s ⁢ ( ℝ N ) ∖ { 0 } , L a ′ ⁢ ( U ) = 0 }

the least energy for La. As an easy consequence of Theorem 3.1 we have

(3.2) E a = C p ⁢ o , a .

In the following result we recall [16, Theorem 1.1 and Theorem 1.3] and [33, Theorem 1.5], which state the regularity and the polynomial decay of solutions of (3.1). This decay is much less slower than the one, exponential, of the local case s=1. An explicit proof, which underlines some uniformity in a, can be found in the Appendix (see Proposition A.4).

Proposition 3.2 (Regularity and Polynomial Decay).

Assume (f1) with respect to a>0 and (f2). Then every weak solution U∈Hs⁢(RN) of (3.1) is actually a strong solution, i.e., U satisfies (3.1) almost everywhere. Moreover,

U ∈ H 2 ⁢ s ⁢ ( ℝ N ) ∩ C σ ⁢ ( ℝ N )

for every σ∈(0,2⁢s), and

U > 0 𝑎𝑛𝑑 lim | x | → + ∞ ⁡ U ⁢ ( x ) = 0 .

Furthermore, we have a polynomial decay at infinity, that is, there exist positive constants Ca′,Ca′′ such that

(3.3) C a ′ 1 + | x | N + 2 ⁢ s ≤ U ⁢ ( x ) ≤ C a ′′ 1 + | x | N + 2 ⁢ s for x ∈ ℝ N .

We observe that the bounding functions in (3.3) belong to Lq⁢(ℝN) for any q∈[1,+∞].

We end this section by a technical lemma, which allows to link the level La⁢(u) of a whatever function u, having a functional Pa⁢(u)≈1, with the ground state Ea; in particular, it provides a useful lower bound for La.

Lemma 3.3.

Let u∈Hs⁢(RN) and define

g ⁢ ( t ) := 1 2 ⁢ s ⁢ ( N ⁢ t N - 2 ⁢ s - ( N - 2 ⁢ s ) ⁢ t N ) , t ∈ ℝ .

If P a ⁢ ( u ) ∈ ( 0 , ( N N - 2 ⁢ s ) 1 2 ⁢ s ) , which we highlight is a neighborhood of 1 , then

L a ⁢ ( u ) ≥ g ⁢ ( P a ⁢ ( u ) ) ⁢ E a .
If u = U a ⁢ ( ⋅ - q t ) for some q ∈ ℝ N and t ∈ ℝ , with U a being a ground state of ( 3.1 ), then the above inequality is indeed an equality, that is,

L a ⁢ ( U a ⁢ ( ⋅ - q t ) ) = g ⁢ ( t ) ⁢ E a .

We highlight that the function g verifies

g ⁢ ( t ) ≤ 1 𝑎𝑛𝑑 g ⁢ ( t ) = 1 ⇔ t = 1 .

Proof.

Let σ:=Pa⁢(u) and set v:=u(σ⋅). Then Pa⁢(v)=1. A straightforward computation shows, by using that Pa⁢(v)=1 and that g⁢(σ)>0,

L a ⁢ ( u ) = g ⁢ ( σ ) ⁢ L a ⁢ ( v ) ≥ g ⁢ ( σ ) ⁢ C p ⁢ o , a = g ⁢ ( σ ) ⁢ E a .

We see that, if u=Ua⁢(⋅-qt), then by Pa⁢(Ua)=1 we have σ=Pa⁢(Ua⁢(⋅-qt))=t and thus v=Ua(⋅-q), which by translation invariance is again a ground state of (3.1); thus La⁢(v)=Cp⁢o,a. This concludes the proof. ∎

3.2 A Family of Equations

In this subsection we study equation (3.1) for variable values of a>0. Introduce the notation

Ω ⁢ [ a , b ] := V - 1 ⁢ ( [ m 0 + a , m 0 + b ] ) ∩ Ω

and similarly Ω⁢(a,b) and mixed-brackets combinations. We choose now a small ν0>0 such that the minimum m0 is not heavily perturbed, namely

Berestycki–Lions assumptions (f1) hold with respect to a∈[m0,m0+ν0] – i.e., in particular, we have F⁢(t0)>12⁢(m0+ν0)⁢t02,
assumption (2) holds with respect to m0+ν0, i.e., m0+ν0<inf∂⁡Ω⁡V,
Ω ⁢ [ 0 , ν 0 ] ¯ ⊂ K d ⊂ Ω for a sufficiently small d>0 subsequently fixed, see Lemma 5.4,
other conditions subsequently stressed, see e.g. (3.4) and Lemma 3.5.

We observe that, by construction, for a∈[m0,m0+ν0] the considerations of Section 3.1 apply. Moreover, by scaling arguments on Cp⁢o,a, we notice that a∈[m0,m0+ν0]↦Ea∈(0,+∞) is strictly increasing and that, up to choosing a smaller ν0, we have Em0+ν0<2⁢Em0 and thus we can find an l0=l0⁢(ν0)∈ℝ such that

(3.4) E m 0 + ν 0 < l 0 < 2 ⁢ E m 0 .

At the end of the paper, we will make ν0 and l0 moving such that ν0n→0 and l0n→Em0. We now define the set of almost ground states of (3.1)

S a := { U ∈ H s ⁢ ( ℝ N ) ∖ { 0 } | L a ′ ⁢ ( U ) = 0 , L a ⁢ ( U ) ≤ l 0 , U ⁢ ( 0 ) = max ℝ N ⁡ U } ≠ ∅ .

We observe that we set the last condition in order to fix solutions in a point and prevent them to escape to infinity; the idea is to gain thickness and compactness (see [14, 23, 21]). We further define

S ^ := ⋃ a ∈ [ m 0 , m 0 + ν 0 ] S a .

The following properties of the set S^ will be of key importance in the whole paper.

Lemma 3.4.

The following properties hold.

There exist positive constants C ′ , C ′′ such that, for each U ∈ S ^ , we have

(3.5) C ′ 1 + | x | N + 2 ⁢ s ≤ U ⁢ ( x ) ≤ C ′′ 1 + | x | N + 2 ⁢ s for x ∈ ℝ N .
S ^ is compact. Since it does not contain the zero function, we have

r * := min U ∈ S ^ ∥ U ∥ H s ⁢ ( ℝ N ) > 0 ;

the maximum is attained as well.
We have

lim R → + ∞ ⁡ ∥ U ∥ ℝ N ∖ B R = 0 , uniformly for U ∈ S ^ ,

where the norm ∥ ⋅ ∥ ℝ N ∖ B R is defined in ( 2.3 ). Moreover, if ( U n ) n ⊂ S ^ , and ( θ n ) n ⊂ ℝ N is included in a compact set, then

lim n → + ∞ ∥ U n ( ⋅ + θ n ) ∥ ℝ N ∖ B n = 0 .

Proof.

We divide the proof into five steps.

Step 1. We see that S^ is bounded. Indeed, by the Pohozaev identity, we have

∥ ( - Δ ) s 2 ⁢ U ∥ 2 2 = N s ⁢ L a ⁢ ( U ) ≤ N s ⁢ l 0 .

By (2.2) we have that also ∥U∥2s* is uniformly bounded. Since La′⁢(U)⁢U=0, we have by (2.5)

∥ ( - Δ ) s 2 ⁢ U ∥ 2 2 + a ⁢ ∥ U ∥ 2 2 = ∫ ℝ N f ⁢ ( U ) ⁢ U ⁢ 𝑑 x ≤ β ⁢ ∥ U ∥ 2 2 + C β ⁢ ∥ U ∥ 2 s * 2 s *

which implies, by choosing β<a, that also ∥U∥22 is bounded.

Step 2. There exist uniform C>0 and σ∈(0,1) such that

(3.6) ∥ U ∥ ∞ ≤ C , [ U ] C loc 0 , σ ⁢ ( ℝ N ) ≤ C

for any U∈S^. We postpone to the Appendix the proof of (3.6), as well as the proof of the uniform pointwise estimate (3.5); see Proposition A.4.

We show now (b), which is an adaptation of the fact that Sa itself is compact.

Step 3. We observe first that S^ is closed. Indeed, if Uk∈Sak⊂S^ converges strongly to U, then up to a subsequence we have ak→a∈[m0,m0+ν0] and, by the strong convergence, we have that the condition

E m 0 ≤ L a ⁢ ( U ) ≤ l 0

holds, which in particular implies that U≢0. Moreover, exploiting the weak convergence Uk⇀U, we obtain by classical arguments that for each v∈Hs⁢(ℝN)

0 = L a k ′ ⁢ ( U k ) ⁢ v = ∫ ℝ N ( - Δ ) s 2 ⁢ U k ⁢ ( - Δ ) s 2 ⁢ v ⁢ 𝑑 x + a k ⁢ ∫ ℝ N U k ⁢ v ⁢ 𝑑 x - ∫ ℝ N f ⁢ ( U k ) ⁢ v ⁢ 𝑑 x

→ ∫ ℝ N ( - Δ ) s 2 ⁢ U ⁢ ( - Δ ) s 2 ⁢ v ⁢ 𝑑 x + a ⁢ ∫ ℝ N U ⁢ v ⁢ 𝑑 x - ∫ ℝ N f ⁢ ( U ) ⁢ v ⁢ 𝑑 x = L a ′ ⁢ ( U ) ⁢ v ,

that is, La′⁢(U)=0. As regards the maximum in zero, we need a pointwise convergence. In order to get it, we exploit the fact that, by (3.6), Uk are uniformly bounded in L∞⁢(ℝN) and in Cloc0,σ⁢(ℝN) and we apply the Ascoli–Arzelà theorem to get local uniform convergence. This shows that U∈Sa⊂S^.

Step 4. Let now Uk∈Sak⊂S^. By the boundedness, up to a subsequence we have Uk⇀U∈Hs⁢(ℝN) and ak→a∈[m0,m0+ν0]. We need to show that ∥Uk∥Hs⁢(ℝN)→∥U∥Hs⁢(ℝN); the closedness of S^ will conclude the proof. To this end, as observed in Step 3, we have by the weak convergence that Lak′⁢(Uk)⁢Uk=0=La′⁢(U)⁢U; hence, if R>0 is a radius to be fixed, we gain

| ( ∥ ( - Δ ) s 2 U k ∥ 2 2 + a k ∥ U k ∥ 2 2 ) - ( ∥ ( - Δ ) s 2 U ∥ 2 2 + a ∥ U ∥ 2 2 ) |

≤ | ∫ | x | ≤ R f ( U k ) U k d x - ∫ | x | ≤ R f ( U ) U d x | + ∫ | x | > R | f ( U k ) U k | d x + ∫ | x | > R | f ( U ) U ) | d x = : ( I ) + ( II ) .

Fix now a small η>0. As regards (II), we have by (2.5) (recall that the definition of p is given in condition f1.3)

∫ | x | > R | f ⁢ ( U k ) ⁢ U k | ⁢ 𝑑 x ≤ β ⁢ ∫ | x | > R | U k | 2 ⁢ 𝑑 x + C β ⁢ ∫ | x | > R | U k | p + 1 ⁢ 𝑑 x < η

for sufficiently (uniformly in k) large R>0 thanks to (3.5); up to taking a larger R, it holds also for U. Fixed this R>0, focusing on (I), by classical arguments we have

| ∫ | x | ≤ R f ⁢ ( U k ) ⁢ U k ⁢ 𝑑 x - ∫ | x | ≤ R f ⁢ ( U ) ⁢ U ⁢ 𝑑 x | < η

for sufficiently large k. Merging together, we obtain

∥ ( - Δ ) s 2 ⁢ U k ∥ 2 2 + a k ⁢ ∥ U k ∥ 2 2 → ∥ ( - Δ ) s 2 ⁢ U ∥ 2 2 + a ⁢ ∥ U ∥ 2 2

which with elementary passages leads to the claim.

Step 5. Finally, we prove (c). By contradiction, there exists an η>0 such that, for each n∈ℕ, there exists Un∈S^ which satisfies ∥Un∥ℝN∖Bn>η. By the compactness, we have, up to a subsequence, Un→U∈S^ as n→+∞. Thus (notice that ∫ℝN|U⁢(x)-U⁢(y)|2|x-y|N+2⁢s⁢𝑑x∈L1⁢(ℝN) and absolute integrability of the integral applies)

η < ∥ U n ∥ ℝ N ∖ B n ≤ ∥ U n - U ∥ H s ⁢ ( ℝ N ) + ∥ U ∥ ℝ N ∖ B n → 0

which is an absurd. For the second part, we argue similarly. Indeed, up to a subsequence, Un→U in Hs⁢(ℝN) and θn→θ in ℝN, thus

∥ U n ( ⋅ - θ n ) ∥ ℝ N ∖ B n ≤ ∥ U n - U ∥ H s ⁢ ( ℝ N ) + ∥ τ θ n U - τ θ U ∥ H s ⁢ ( ℝ N ) + ∥ U ( ⋅ - θ ) ∥ ℝ N ∖ B n → 0 ,

where τθ is the translation. This concludes the proof. ∎

Gained compactness, we turn back considering the set of all the solutions (with no restrictions in zero), that is,

S ^ ′ := ⋃ p ∈ ℝ N τ p ⁢ ( S ^ ) ;

we observe that S^′ is bounded. Moreover, we define an open r-neighborhood of S^′, reminiscent of the perturbation approach in [35, 3, 26],

S ⁢ ( r ) := { u ∈ H s ⁢ ( ℝ N ) ∣ d ⁢ ( u , S ^ ′ ) < r } ,

that is,

S ( r ) = { u = U ( ⋅ - p ) + φ ∈ H s ( ℝ N ) ∣ U ∈ S ^ , p ∈ ℝ N , φ ∈ H s ( ℝ N ) , ∥ φ ∥ H s ⁢ ( ℝ N ) < r } .

We further define a minimal radius map ρ^:Hs⁢(ℝN)→ℝ+ by

ρ ^ ( u ) := inf { ∥ u - U ( ⋅ - y ) ∥ H s ⁢ ( ℝ N ) ∣ U ∈ S ^ , y ∈ ℝ N } .

We observe

(3.7) u ∈ S ⁢ ( r ) ⟹ ρ ^ ⁢ ( u ) < r ,

and in addition

ρ ^ ⁢ ( u ) = inf ⁡ { t ∈ ℝ + ∣ u ∈ S ⁢ ( t ) } ,

where the infimum on the right-hand side is not attained. Finally, ρ^∈Lip⁢(Hs⁢(ℝN),ℝ) with Lipschitz constant equal to 1, that is, for every u,v∈Hs⁢(ℝN),

(3.8) | ρ ^ ⁢ ( u ) - ρ ^ ⁢ ( v ) | ≤ ∥ u - v ∥ H s ⁢ ( ℝ N ) .

We end this subsection with two technical lemmas. The first one is a direct consequence of Lemma 3.3, and allows to link the level Lm0⁢(u) of a whatever function u∈S⁢(r) with the ground state Em0, once r is sufficiently small; this further gives a lower bound for the functional Lm0.

Lemma 3.5.

Up to taking a smaller ν0=ν0⁢(l0)>0, there exists a sufficiently small r′=r′⁢(ν0,r*)>0 such that, for every u∈S⁢(r′), we have

L m 0 ⁢ ( u ) ≥ g ⁢ ( P m 0 ⁢ ( u ) ) ⁢ E m 0 .

Proof.

By Lemma 3.3 (a), we know that the inequality holds if Pm0⁢(u) is in a neighborhood of the value 1. Observe that Pa⁢(U)=1 if U∈Sa: by continuity and compactness, Pm0⁢(U)≈1 if U∈Sa and a≈m0. In particular, by choosing a small value of ν0, Pm0⁢(U)≈1 for U∈S^. Indeed,

P m 0 ⁢ ( U ) = 1 + 1 2 ⁢ s ⁢ N N - 2 ⁢ s ⁢ ( a - m 0 ) ⁢ ∥ U ∥ 2 2 ∥ ( - Δ ) s 2 ⁢ U ∥ 2 2 + o ⁢ ( 1 ) ;

the addendum on the right-hand side can be bounded by the maximum and the minimum over S^ (notice that (-Δ)s2⁢U cannot be zero) and thus we can find a uniform small ν0. Again by continuity we have that Pm0⁢(u)≈1 for u∈S⁢(r′), r′ is sufficiently small. Indeed

P m 0 ( u ) = 1 - 1 2 ⁢ s N N - 2 ⁢ s 1 ∥ ( - Δ ) s 2 ⁢ U ∥ 2 2 ( L m 0 ( U ( ⋅ - p ) + φ ) - L m 0 ( U ( ⋅ - p ) ) ) + o ( 1 ) .

This concludes the proof. ∎

We notice that the condition Em0+ν0<l0 keeps holding by decreasing ν0, so no ambiguity in the l0-depending choice of ν0 in Lemma 3.5 arises. We focus now on the second lemma.

Lemma 3.6.

There exist ν1∈(0,ν0) and δ0=δ0⁢(ν1)>0 such that

L m 0 + ν 1 ⁢ ( U ) ≥ E m 0 + δ 0 , uniformly for U ∈ S ^ .

Proof.

Observe first that, since S^ is compact also in L2⁢(ℝN), we have also finite and strictly positive minimum M¯ and maximum M¯ with respect to ∥⋅∥2. Consider ν1∈(0,ν0) such that

E m 0 + ν 1 - E m 0 > 1 2 ⁢ ( ν 0 - ν 1 ) ⁢ M ¯ ;

we notice that such ν1 exists since, as ν1→ν0+, the left-hand side positively increases while the right-hand side goes to zero. Let now a∈[m0,m0+ν0]; we consider two cases. If a∈[m0,m0+ν1], we argue as follow: for U∈Sa we have

L m 0 + ν 1 ( U ) = L a ( U ) + 1 2 ( m 0 + ν 1 - a ) ∥ U ∥ 2 2 ≥ E a + 1 2 ( m 0 + ν 1 - a ) M ¯ = : ( I ) + ( II ) ;

now, the quantity (I) is minimum when a=m0, while (II) is minimum when a=m0+ν1; if both could apply at the same time, we would have as a minimum the quantity Em0. Since it is not possible, we obtain

inf U ∈ ⋃ a ∈ [ m 0 , m 0 + ν 1 ] S a ⁡ L m 0 + ν 1 ⁢ ( U ) > E m 0 .

If a∈(m0+ν1,m0+ν0] instead, we have

L m 0 + ν 1 ⁢ ( U ) ≥ E a - 1 2 ⁢ ( a - ( m 0 + ν 1 ) ) ⁢ M ¯ ≥ E m 0 + ν 1 - 1 2 ⁢ ( ν 0 - ν 1 ) ⁢ M ¯

and thus by the assumption

inf U ∈ ⋃ a ∈ [ m 0 + ν 1 , m 0 + ν 0 ] S a ⁡ L m 0 + ν 1 ⁢ ( U ) ≥ E m 0 + ν 1 - 1 2 ⁢ ( ν 0 - ν 1 ) ⁢ M ¯ > E m 0 .

This concludes the proof, by taking as δ0>0 the smallest of the two differences. ∎

3.3 Fractional Center of Mass

As in [21], inspired by the works [10, 28, 17], we want to define a barycentric map Υ which, given a function u=U(⋅-p)+φ∈S(r), gives an estimate on the maximum point p of U(⋅-p); since φ is small, p is, in some ways, the center of mass of u. The idea will be to bound Υ⁢(u) in order to re-gain compactness.

Since the nonlocality comes into the very definition of the ambient space, we need the use of the norm (2.3), which we notice being stronger than the one induced by the Gagliardo seminorm.

Lemma 3.7.

Let r* be as in Lemma 3.4. Then there exist a sufficiently large R0>0, a sufficiently small radius r0∈(0,r*) and a continuous map

Υ : S ⁢ ( r 0 ) → ℝ N

such that, for each u=U(⋅-p)+φ∈S(r0), we have

| Υ ⁢ ( u ) - p | ≤ 2 ⁢ R 0 .

Moreover, Υ is continuous and -Υ is shift-equivariant, that is, Υ(u(⋅+ξ))=Υ(u)-ξ for every u∈S⁢(r0) and ξ∈RN.

Proof.

Recalled that r*=minU∈S^∥U∥Hs⁢(ℝN)>0, we have by Lemma 3.4,

(3.9) ∥ U ∥ ℝ N ∖ B R 0 < 1 8 ⁢ r * , uniformly for U ∈ S ^ ,

for R0≫0. Thus

r * ≤ ∥ U ∥ H s ⁢ ( ℝ N ) ≤ ∥ U ∥ B R 0 + ∥ U ∥ ℝ N ∖ B R 0 < ∥ U ∥ B R 0 + 1 8 ⁢ r * ,

which implies

∥ U ∥ B R 0 > 7 8 ⁢ r * and ∥ U ∥ ℝ N ∖ B R 0 < 1 8 ⁢ r *

for each U∈S^. Consider now a cutoff function ψ∈Cc∞⁢(ℝ+) such that

[ 0 , 1 4 ⁢ r * ] ≺ ψ ≺ [ 1 2 ⁢ r * , + ∞ ) ;

let r0∈(0,18⁢r*) and define, for each u∈S⁢(r0) and q∈ℝN, a density function

d ( q , u ) := ψ ( inf U ~ ∈ S ^ ∥ u - U ~ ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) ) .

Notice that d⁢(⋅,u) is an integrable function: indeed, we have [u(⋅+ξ)]BR0⁢(q)=[u]BR0⁢(q+ξ) and the function given by q↦∥u-U~(⋅-q)∥BR0⁢(q)=∥τqu-U~∥BR0 is continuous by

| ∥ τ q ⁢ u - U ~ ∥ B R 0 - ∥ τ p ⁢ u - U ~ ∥ B R 0 | ≤ ∥ τ q ⁢ u - τ p ⁢ u ∥ B R 0 ≤ ∥ τ q ⁢ u - τ p ⁢ u ∥ H s ⁢ ( ℝ N ) → 0

as p→q, and the infimum over continuous functions is upper semicontinuous.

If we show that d⁢(q,u)≥0 is not identically zero and it has compact support, then it will be well defined the quantity

Υ ⁢ ( u ) := ∫ ℝ N q ⁢ d ⁢ ( q , u ) ⁢ 𝑑 q ∫ ℝ N d ⁢ ( q , u ) ⁢ 𝑑 q .

We show first that d⁢(⋅,u) has compact support. Indeed if u=U(⋅-p)+φ and U~∈S^ is arbitrary, then

∥ u - U ~ ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) ≥ ∥ U ~ ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) - ∥ U ( ⋅ - p ) ∥ B R 0 ⁢ ( q ) - ∥ φ ∥ B R 0 ⁢ ( q )

≥ ∥ U ~ ∥ B R 0 - ∥ U ∥ B R 0 ⁢ ( q - p ) - ∥ φ ∥ H s ⁢ ( ℝ N )

≥ 6 8 ⁢ r * - ∥ U ∥ B R 0 ⁢ ( q - p ) ;

take now q∉B2⁢R0⁢(p): if x∈BR0⁢(q-p), by the fact that |x-(q-p)|<R0 and |q-p|≥2⁢R0, we obtain that |x|≥R0, that is, x∈ℝN∖BR0. Therefore by (3.9)

(3.10) ∥ u - U ~ ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) ≥ 6 8 r * - ∥ U ∥ ℝ N ∖ B R 0 ≥ 5 8 r * > 1 2 r *

thus

inf U ∈ S ^ ∥ u - U ~ ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) ≥ 1 2 r *

and hence d⁢(q,u)=0 for q∉B2⁢R0⁢(p); this means that supp(d⁢(⋅,u))⊂B2⁢R0⁢(p).

We show next that d⁢(⋅,u) is equal to 1 on a ball. Indeed if u=U(⋅-p)+φ

inf U ~ ∈ S ^ ∥ u - U ~ ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) ≤ ∥ u - U ( ⋅ - q ) ∥ B R 0 ⁢ ( q )

≤ ∥ U ( ⋅ - p ) - U ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) + ∥ φ ∥ B R 0 ⁢ ( q )

≤ ∥ τ p - q ⁢ U - U ∥ B R 0 + 1 8 ⁢ r * .

We can make the first term as small as we want by taking |p-q| small, that is,

inf U ∈ S ^ ∥ u - U ~ ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) ≤ 1 4 r *

for q∈Br⁢(p), r small, which implies d⁢(q,u)=1.

By the fact that Br⁢(p)⊂supp(d⁢(⋅,u))⊂B2⁢R0⁢(p) we have the well posedness of Υ⁢(u) and

Υ ⁢ ( u ) = ∫ B 2 ⁢ R 0 ⁢ ( p ) q ⁢ d ⁢ ( q , u ) ⁢ 𝑑 q ∫ B 2 ⁢ R 0 ⁢ ( p ) d ⁢ ( q , u ) ⁢ 𝑑 q .

The main property comes straightforward, as well as the shift equivariance. We show now the continuity. Indeed, assume ∥u-v∥Hs⁢(ℝN)≤18⁢r*. Then, by (3.10),

∥ v - U ~ ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) ≥ ∥ u - U ~ ( ⋅ - q ) ∥ B R 0 ⁢ ( q ) - ∥ v - u ∥ B R 0 ⁢ ( q ) ≥ 1 2 r *

and again we can conclude that supp(d⁢(⋅,v))⊂B2⁢R0⁢(p) for each ∥u-v∥Hs⁢(ℝN)≤18⁢r*, where p depends only on u. Moreover, observe that

∫ B 2 ⁢ R 0 ⁢ ( p ) d ( q , u ) d q ≥ ∫ B r ⁢ ( p ) 1 d q ≥ | B r | = : C 1

not depending on u and p (and similarly C2:=|B2⁢R0|), and that d⁢(q,⋅) is Lipschitz (since ψ and the norm are so, and the infimum over a family of Lipschitz function is still Lipschitz). Thus we have

Υ ⁢ ( u ) - Υ ⁢ ( v ) ≤ ∫ B 2 ⁢ R 0 ⁢ ( p ) | q | ⁢ | d ⁢ ( q , u ) - d ⁢ ( q , v ) | ⁢ 𝑑 q ∫ B 2 ⁢ R 0 ⁢ ( p ) d ⁢ ( q , u ) ⁢ 𝑑 q

+ ∫ B 2 ⁢ R 0 ⁢ ( p ) | q | ⁢ d ⁢ ( q , v ) ⁢ 𝑑 q ⁢ ∫ B 2 ⁢ R 0 ⁢ ( p ) | d ⁢ ( q , v ) - d ⁢ ( q , u ) | ⁢ 𝑑 q ∫ B 2 ⁢ R 0 ⁢ ( p ) d ⁢ ( q , u ) ⁢ 𝑑 q ⁢ ∫ B 2 ⁢ R 0 ⁢ ( p ) d ⁢ ( q , v ) ⁢ 𝑑 q

≤ ∫ B 2 ⁢ R 0 ⁢ ( p ) | q | d q 1 C 1 ( 1 + C 2 C 1 ) ∥ u - v ∥ H s ⁢ ( ℝ N ) = : C p ∥ u - v ∥ H s ⁢ ( ℝ N ) .

Since Cp can be bounded above by a constant of the type C⁢(1+Υ⁢(u)), we have

∥ u - v ∥ H s ⁢ ( ℝ N ) ≤ r 0 ⟹ | Υ ⁢ ( u ) - Υ ⁢ ( v ) | ≤ C ⁢ ( 1 + Υ ⁢ ( u ) ) ⁢ ∥ u - v ∥ H s ⁢ ( ℝ N ) ;

in particular, this implies the continuity. ∎

4 Singularly Perturbed Equation

We come back now to our equation

(4.1) ( - Δ ) s ⁢ u + V ⁢ ( ε ⁢ x ) ⁢ u = f ⁢ ( u ) , x ∈ ℝ N .

It is known that the solutions of equation (4.1) can be characterized as critical points of the functional Iε:Hs⁢(ℝN)→ℝ defined by

I ε ⁢ ( u ) := 1 2 ⁢ ∥ ( - Δ ) s 2 ⁢ u ∥ 2 2 + 1 2 ⁢ ∫ ℝ N V ⁢ ( ε ⁢ x ) ⁢ u 2 ⁢ 𝑑 x - ∫ ℝ N F ⁢ ( u ) ⁢ 𝑑 x , u ∈ H s ⁢ ( ℝ N ) ,

where Iε∈C1⁢(Hs⁢(ℝN),ℝ), since ∥⋅∥Hεs⁢(ℝN) is a norm.

We start with a technical result. Let ν1 be as in Lemma 3.6; we want to show that the claim of the lemma continues holding, for ε small, if we replace Lm0+ν1 with Iε, and S^ with S(r0′)∩{εΥ(u)∈Ω[ν1,ν0]}, r0′ small.

Lemma 4.1.

Let ν1 and δ0 be as in Lemma 3.6. Then there exist δ1∈(0,δ0) and r0′=r0′⁢(δ1)∈(0,r0) sufficiently small such that, for every ε small, we have

I ε ⁢ ( u ) ≥ E m 0 + δ 1

for each u∈{u∈S⁢(r0′)∣ε⁢Υ⁢(u)∈Ω⁢[ν1,+∞)}⊃{u∈S⁢(r0′)∣ε⁢Υ⁢(u)∈Ω⁢[ν1,ν0]}.

Proof.

First we improve Lemma 3.6 for La in the direction of the nonautonomous equation. Indeed, by the assumption, we have V⁢(ε⁢Υ⁢(u))≥m0+ν1, that is,

L V ⁢ ( ε ⁢ Υ ⁢ ( u ) ) ⁢ ( U ) ≥ L m 0 + ν 1 ⁢ ( U ) ≥ E m 0 + δ 0

for any U∈S^. Moreover, if u=U~(⋅-p)+φ~∈S(r0) then, by Lemma 3.7, ε⁢p∈Ω2⁢ε⁢R0⊂Ω2⁢R0 which is compact. By uniform continuity of V and boundedness from above of S^, we have

(4.2) L V ⁢ ( ε ⁢ p ) ⁢ ( U ) ≥ E m 0 + δ 0 / 2

for all U∈S^ and ε small enough.

Let now r0′ to be fixed and u=U(⋅-p)+φ∈S(r1). Then we have

I ε ( u ) = I ε ( U ( ⋅ - p ) + φ ) = I ε ( U ( ⋅ - p ) ) + I ε ′ ( v ) φ

for some v∈Hs⁢(ℝN) in the segment [U(⋅-p),u]. Notice that v lies in a ball of radius max⁡S^+r0′ and Iε′ sends bounded sets in bounded sets (uniformly on ε); thus there exists a constant C, not depending on U, p and φ, such that

(4.3) I ε ( u ) ≥ I ε ( U ( ⋅ - p ) ) - C ∥ φ ∥ H s ⁢ ( ℝ N ) ≥ I ε ( U ( ⋅ - p ) ) - δ 1 2

for ∥φ∥Hs⁢(ℝN)<r0′ sufficiently small. Recalled that ε⁢p∈Ω2⁢R0 we have, by the uniform continuity of V and the uniform estimate (3.5), for sufficiently small ε,

I ε ( U ( ⋅ - p ) ) ≥ L V ⁢ ( ε ⁢ p ) ( U ) - δ 1 2

and the claim comes from (4.3) and (4.2), since, for δ1<δ04,

I ε ⁢ ( u ) ≥ E m 0 + δ 0 2 - δ 1 ≥ E m 0 + δ 1 . ∎

Before introducing the penalized functional, we state another technical lemma, which gives a (trivial, but useful) lower bound for Iε′⁢(v) for small values of v.

Lemma 4.2.

There exists r1>0 sufficiently small (see also the Remark below) and a constant C>0 such that

(4.4) I ε ′ ⁢ ( v ) ⁢ v ≥ C ⁢ ∥ v ∥ H s ⁢ ( ℝ N ) 2

for every ε>0 and v∈Hs⁢(RN) with ∥v∥Hs⁢(RN)≤r1.

Proof.

We have, by (2.5) with β<12⁢V¯,

I ε ′ ⁢ ( v ) ⁢ v ≥ ∥ ( - Δ ) s 2 ⁢ v ∥ 2 2 + ∫ ℝ N V ¯ ⁢ v 2 ⁢ 𝑑 x - β ⁢ ∥ v ∥ 2 2 - C β ⁢ ∥ v ∥ p + 1 p + 1

(4.5) ≥ ∥ ( - Δ ) s 2 ⁢ v ∥ 2 2 + 1 2 ⁢ V ¯ ⁢ ∥ v ∥ 2 2 - C β ⁢ ∥ v ∥ p + 1 p + 1 ≥ C ⁢ ∥ v ∥ H s ⁢ ( ℝ N ) 2 - C β ⁢ ∥ v ∥ H s ⁢ ( ℝ N ) p + 1 ≥ C ′ ⁢ ∥ v ∥ H s ⁢ ( ℝ N ) 2 ,

where the last inequality holds for ∥v∥Hs⁢(ℝN) small, since p+1>2. ∎

Remark 4.3.

For a later use, we observe that one can improve (4.5) by

(4.6) ∥ ( - Δ ) s 2 ⁢ v ∥ 2 2 + 1 2 ⁢ V ¯ ⁢ ∥ v ∥ 2 2 - 2 p ⁢ C β ⁢ ∥ v ∥ p + 1 p + 1 ≥ C ⁢ ∥ v ∥ H s ⁢ ( ℝ N ) 2

up to choosing a smaller r1.

4.1 A Penalized Functional

We want to study now a penalized functional (see [21]), that is, Iε plus a term which forces solutions to stay in Ω; we highlight that if we had information on V at infinity (for example, trapping potentials) we would not have need of this penalization.

Since V>m0 on ∂⁡Ω, we can find an annulus around ∂⁡Ω where this relation keeps holding, that is,

V ⁢ ( x ) > m 0 for x ∈ Ω 2 ⁢ h 0 ∖ Ω ¯

for h0 sufficiently small. We then define the penalization (or mass-concentrating) functional Qε:Hs⁢(ℝN)→ℝ

Q ε ⁢ ( u ) := ( 1 ε α ⁢ ∥ u ∥ L 2 ⁢ ( ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) ) 2 - 1 ) + p + 1 2 , u ∈ H s ⁢ ( ℝ N ) ,

where α∈(0,min⁡{12,s}).

We observe that, for every u,v∈Hs⁢(ℝN),

Q ε ′ ⁢ ( u ) ⁢ v = ( p + 1 ) ε α ⁢ ( 1 ε α ⁢ ∥ u ∥ L 2 ⁢ ( ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) ) 2 - 1 ) + p - 1 2 ⁢ ∫ ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) u ⁢ v ⁢ 𝑑 x

and it is straightforward to prove the following estimate:

(4.7) Q ε ′ ⁢ ( u ) ⁢ u ≥ ( p + 1 ) ⁢ Q ε ⁢ ( u ) .

We further set

J ε := I ε + Q ε ,

the penalized functional. It results that Qε and Jε are in C1⁢(Hs⁢(ℝN),ℝ).

We want to find critical points of Jε and show, afterwards, that these critical points, under suitable assumptions, are critical points of Iε too, since Qε will be identically zero. Let ε=1: observed that Q1⁢(u) vanishes if u have much mass inside Ω, we see that J1⁢(u)=I1⁢(u) holds when the mass of uconcentrates in Ω; this is why we say that Qεforcesu to stay in Ω. Similarly, as ε→0, much less mass must be found outside Ω/ε.

We start by two technical lemmas. The first one gives a sufficient condition to pass from weak to strong convergent sequences in a Hilbert space, similarly to the convergence of the norms.

Lemma 4.4.

Fix ε>0 and let (uj)j⊂Hs⁢(RN) be such that

(4.8) ∥ J ε ′ ⁢ ( u j ) ∥ H - s ⁢ ( ℝ N ) → 0 as j → + ∞ .

Assume moreover that uj⇀u0 in Hs⁢(RN) as j→+∞, and that

(4.9) lim R , j → + ∞ ⁡ ∥ u j ∥ L q ⁢ ( ℝ N ∖ B R ) = 0

for q=2 and q=p+1. Then uj→u0 in Hs⁢(RN) as j→+∞.

Proof.

We have by the weak lower semicontinuity of the norm

(4.10) lim inf j → + ∞ ⁡ ∥ u j ∥ H ε s ⁢ ( ℝ N ) ≥ ∥ u 0 ∥ H ε s ⁢ ( ℝ N ) .

Moreover,

∥ u j ∥ H ε s ⁢ ( ℝ N ) 2 = ∫ ℝ N f ⁢ ( u j ) ⁢ u j ⁢ 𝑑 x + I ε ′ ⁢ ( u j ) ⁢ u j

= ( ∫ ℝ N f ⁢ ( u j ) ⁢ u j ⁢ 𝑑 x - ∫ ℝ N f ⁢ ( u 0 ) ⁢ u 0 ⁢ 𝑑 x ) + ( I ε ′ ⁢ ( u j ) ⁢ u j - I ε ′ ⁢ ( u 0 ) ⁢ u 0 ) + I ε ′ ⁢ ( u 0 ) ⁢ u 0 + ∫ ℝ N f ⁢ ( u 0 ) ⁢ u 0 ⁢ 𝑑 x

= : ( I ) + ( II ) + ∥ u 0 ∥ H ε s ⁢ ( ℝ N ) 2 ;

if we prove that

lim sup j → + ∞ ⁡ ( ( I ) + ( II ) ) ≤ 0 ,

we are done, because together with (4.10) we obtain

∥ u j ∥ H ε s ⁢ ( ℝ N ) → ∥ u 0 ∥ H ε s ⁢ ( ℝ N ) as j → + ∞ ,

which implies the claim, since Hεs⁢(ℝN) is a Hilbert space.

Focus on (I); we have

∫ ℝ N ( f ( u j ) u j - f ( u 0 ) u 0 ) d x = ∫ B R ( f ( u j ) u j - f ( u 0 ) u 0 ) d x + ∫ ℝ N ∖ B R ( f ( u j ) u j - f ( u 0 ) u 0 ) d x = : ( I 1 ) + ( I 2 ) .

The piece (I2) can be made small for j and R sufficiently large, by exploiting the estimates on f, assumption (4.9) and the absolute continuity of the Lebesgue integral for u0. For such large R and j, up to taking a larger j, we can make the piece (I1) small by classical arguments.

Focus now on (II); we first observe that by exploiting Hölder inequalities and again classical arguments we have Iε′⁢(uj)⁢u0→Iε′⁢(u0)⁢u0. Thus we have, by (4.8),

lim sup j → + ∞ ⁡ ( I ε ′ ⁢ ( u j ) ⁢ u j - I ε ′ ⁢ ( u 0 ) ⁢ u 0 ) = - lim inf j → + ∞ ⁡ ( Q ε ′ ⁢ ( u j ) ⁢ u j - Q ε ′ ⁢ ( u j ) ⁢ u 0 )

= - lim inf j → + ∞ ( ( 1 ε α ∥ u j ∥ L 2 ⁢ ( ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) ) 2 - 1 ) + p - 1 2

⋅ ( ∫ ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) u j 2 d x - ∫ ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) u j u 0 d x ) ) ≤ 0 ,

where the last inequality is due to the following fact: observe first that

u j ⇀ u 0 in H ε s ⁢ ( ℝ N ) ↪ L 2 ⁢ ( ℝ N ) ,

thus (by restriction) uj⇀u0 in L2⁢(ℝn∖(Ω2⁢h0/ε)); by the definition of weak convergence and by the lower semicontinuity of the norm, we have

lim inf j → + ∞ ⁡ ∫ ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) u j 2 ⁢ 𝑑 x ≥ ∫ ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) u 0 2 ⁢ 𝑑 x = lim j → + ∞ ⁡ ∫ ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) u j ⁢ u 0 ⁢ 𝑑 x ,

that is,

lim inf j → + ∞ ⁡ ( ∫ ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) u j 2 ⁢ 𝑑 x - ∫ ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) u j ⁢ u 0 ⁢ 𝑑 x ) ≥ 0 .

Noticed that an≥0 and lim infn⁡bn≥0 imply lim infn⁡(an⁢bn)≥0, we conclude. ∎

The second lemma is a lower bound for Jε with respect to the functional Lm0. We highlight that in what follows we understand that the case m0=V¯, i.e., m0global minimum, gives rise to a not-perturbed result.

Lemma 4.5.

Set Cmin:=12⁢(m0-V¯)≥0 we have, for ε small and u∈Hs⁢(RN),

J ε ⁢ ( u ) ≥ L m 0 ⁢ ( u ) - C min ⁢ ε α .

Proof.

We have, recalling that m0 is the infimum of V over Ω2⁢h0 and V¯ is the infimum over ℝN,

J ε ⁢ ( u ) = L m 0 ⁢ ( u ) + 1 2 ⁢ ∫ ℝ N ( V ⁢ ( ε ⁢ x ) - m 0 ) ⁢ u 2 ⁢ 𝑑 x + Q ε ⁢ ( u )

≥ L m 0 ⁢ ( u ) + 1 2 ⁢ ∫ ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) ( V ⁢ ( ε ⁢ x ) - m 0 ) ⁢ u 2 ⁢ 𝑑 x + Q ε ⁢ ( u )

≥ L m 0 ⁢ ( u ) - C min ⁢ ∥ u ∥ L 2 ⁢ ( ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) ) 2 + Q ε ⁢ ( u ) .

If ∥u∥L2⁢(ℝN∖(Ω2⁢h0/ε))2≤2⁢εα, we have the claim by the positivity of Qε⁢(u). If instead ∥u∥L2⁢(ℝN∖(Ω2⁢h0/ε))2≥2⁢εα, then

Q ε ⁢ ( u ) ≥ ( 1 2 ⁢ ε α ⁢ ∥ u ∥ L 2 ⁢ ( ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) ) 2 ) p + 1 2 ≥ 1 2 ⁢ ε α ⁢ ∥ u ∥ L 2 ⁢ ( ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) ) 2 ,

where we used also that p+12≥1. Thus

J ε ⁢ ( u ) ≥ L m 0 + ( 1 2 ⁢ ε α - C min ) ⁢ ∥ u ∥ L 2 ⁢ ( ℝ N ∖ ( Ω 2 ⁢ h 0 / ε ) ) 2 ≥ L m 0

for ε small. This concludes the proof. ∎

4.2 Critical Points and Palais–Smale Condition

In order to get critical points of Jε we want to implement a deformation argument. As usual, we need a uniform estimate from below of ∥Jε′⁢(u)∥H-s⁢(ℝN), and this is the next goal.

First, by the strict monotonicity of Ea, let us fix l0′=l0′⁢(ν1)>0 such that

E m 0 < l 0 ′ < E m 0 + ν 1 ;

as well as ν0 and l0, even l0′ will be let vary as (l0′)n→Em0 in the proof of the existence.

Lemma 4.6.

Let r0 and r1 be as in Lemma 3.7 and Lemma 4.2. There exists r2′∈(0,min⁡{r0,r1}) sufficiently small with the following property: let 0<ρ1<ρ0≤r2′ and (uε)ε⊂S⁢(r2′) be such that

(4.11) ∥ J ε ′ ⁢ ( u ε ) ∥ H - s ⁢ ( ℝ N ) → 0 ⁢ ⁢ as ε → 0 ,

(4.12) J ε ⁢ ( u ε ) ≤ l 0 ′ < E m 0 + ν 1 ⁢ ⁢ for any ε > 0 ,

with the additional assumption

( ρ ^ ⁢ ( u ε ) ) ε ⊂ [ 0 , ρ 0 ] , ( ε ⁢ Υ ⁢ ( u ε ) ) ε ⊂ Ω ⁢ [ 0 , ν 0 ] .

Then, for ε small,

ρ ^ ⁢ ( u ε ) ∈ [ 0 , ρ 1 ] , ε ⁢ Υ ⁢ ( u ε ) ∈ Ω ⁢ [ 0 , ν 1 ] .

We notice, by (4.11) and (4.12), that (uε)ε resembles a particular (truncated) Palais–Smale sequence. As an immediate consequence of the lemma, set the sublevel

J ε c := { u ∈ H s ⁢ ( ℝ N ) ∣ J ε ⁢ ( u ) ≤ c } ,

we have the following theorem.

Theorem 4.7.

There exists r2′∈(0,min⁡{r0,r1}) sufficiently small with the following property: if 0<ρ1<ρ0≤r2′, then there exists a δ2=δ2⁢(ρ0,ρ1)>0 such that, for ε small,

∥ J ε ′ ⁢ ( u ) ∥ H - s ⁢ ( ℝ N ) ≥ δ 2

for any

u ∈ { u ∈ S ⁢ ( r 2 ′ ) ∩ J ε l 0 ′ ∣ ( ρ ^ ⁢ ( u ) , ε ⁢ Υ ⁢ ( u ) ) ∈ ( ( 0 , ρ 0 ] × Ω ⁢ [ 0 , ν 0 ] ) ∖ ( [ 0 , ρ 1 ] × Ω ⁢ [ 0 , ν 1 ] ) }

⊃ { u ∈ S ⁢ ( r 2 ′ ) ∩ J ε l 0 ′ ∣ ρ 1 < ρ ^ ⁢ ( u ) ≤ ρ 0 , ε ⁢ Υ ⁢ ( u ) ∈ Ω ⁢ ( ν 1 , ν 0 ] } .

Remark 4.8.

Arguing as in the last part of the proof of Lemma 3.4, noticed that S^ is compact not only in Hs⁢(ℝN) but also in Lq⁢(ℝN) for q∈[2,2s*], if (Un)n⊂S^ and (θn)n⊂ℝN is included in a compact, we have

where the norm |||⋅|||ℝN∖Bn is defined in (2.4).

Proof of Lemma 4.6.

We use the notation, for h>0,

Ω h ε := ( Ω ε ⁢ h ) / ε = ( Ω / ε ) h

and notice that if h<h′ then Ω/ε⊂Ωhε⊂Ωh′ε. Let r2′<min⁡{r0,r1} to be fixed.

Step 1. An Estimate for uε. We have, for uε=Uε(⋅-pε)+φε∈S(r2′),

By the fact that ε⁢Υ⁢(uε)∈Ω⁢[ν1,ν0]⊂Ω, we have that 0∈Ω/ε-Υ⁢(uε) and thus Ω/ε-Υ⁢(uε) expands in ℝN as ε→0. Moreover, by Lemma 3.7 we have θε:=Υ⁢(uε)-pε∈B2⁢R0 compact. By Remark 4.8, for ε small we have

(4.13) || | u ε | || ℝ N ∖ ( Ω / ε ) ≤ ( 1 + C ) ⁢ r 2 ′ = C ′ ⁢ r 2 ′

Let

n ε := [ 1 + 4 ⁢ h 0 ε + 1 2 ] ∈ ℕ ,

which by definition satisfies ε⁢nε⁢(nε+1)≤h0 and nε→+∞ as ε→0. We have

∑ i = 1 n ε ∥ u ε ∥ L 2 ⁢ ( Ω n ε ⁢ ( i + 1 ) ε ∖ Ω n ε ⁢ i ε ) 2 ≤ ∥ u ε ∥ L 2 ⁢ ( Ω n ε ⁢ ( n ε + 1 ) ε ∖ Ω n ε ε ) 2 ≤ ( C ′ ⁢ r 2 ′ ) 2

and similarly

∑ i = 1 n ε [ u ε ] Ω n ε ⁢ ( i + 1 ) ε ∖ Ω n ε ⁢ i ε , ℝ N 2 ≤ ( C ′ ⁢ r 2 ′ ) 2 , ∑ i = 1 n ε ∥ u ε ∥ L p + 1 ⁢ ( Ω n ε ⁢ ( i + 1 ) ε ∖ Ω n ε ⁢ i ε ) p + 1 ≤ ( C ′ ⁢ r 2 ′ ) p + 1

thus, for some C=C⁢(r2′),

∑ i = 1 n ε ( ∥ u ε ∥ Ω n ε ⁢ ( i + 1 ) ε ∖ Ω n ε ⁢ i ε 2 + ∥ u ε ∥ L p + 1 ⁢ ( Ω n ε ⁢ ( i + 1 ) ε ∖ Ω n ε ⁢ i ε ) p + 1 ) ≤ C .

This implies that there exists iε∈{1,…,nε} such that

(4.14) ∥ u ε ∥ A ε 2 + ∥ u ε ∥ L p + 1 ⁢ ( A ε ) p + 1 ≤ C n ε → 0 as ε → 0 ,

where

A ε := Ω n ε ⁢ ( i ε + 1 ) ε ∖ Ω n ε ⁢ i ε ε

and C depends on r2′ (we will omit this dependence).

Step 2. Split the Sequence. Consider cutoff functions φε∈Cc∞⁢(ℝN)

Ω n ε ⁢ i ε ε ≺ φ ε ≺ Ω n ε ⁢ ( i ε + 1 ) ε

such that ∥∇⁡φε∥∞≤Cnε=o⁢(1) as ε→0 (which is possible because the distance between Ωnε⁢iεε and Ωnε⁢(iε+1)ε is nε→+∞).

Define

u ε ( 1 ) := φ ε ⁢ u ε , u ε ( 2 ) := ( 1 - φ ε ) ⁢ u ε and u ε := u ε ( 1 ) + u ε ( 2 ) ;

notice that both supp(uε(1)⁢uε(2)) and supp(F⁢(uε)-F⁢(uε(1))-F⁢(uε(2))) are contained in Aε, that is, where we gained the estimate of the norm. Moreover, since

supp ( u ε ( 1 ) ) ⊂ Ω n ε ⁢ ( i ε + 1 ) ε ⊂ Ω ε ⁢ n ε ⁢ ( n ε + 1 ) / ε ⊂ Ω 2 ⁢ h 0 / ε ,

we have, by the definition of Qε, that Qε⁢(uε(1))=0, Qε⁢(uε)=Qε⁢(uε(2)) and

(4.15) Q ε ′ ⁢ ( u ε ( 1 ) ) = 0 , Q ε ′ ⁢ ( u ε ) = Q ε ′ ⁢ ( u ε ( 2 ) ) .

Step 3. Relations of the Functionals. We show that

| I ε ⁢ ( u ε ) - I ε ⁢ ( u ε ( 1 ) ) - I ε ⁢ ( u ε ( 2 ) ) | → 0 as ε → 0

from which

(4.16) J ε ⁢ ( u ε ) = I ε ⁢ ( u ε ( 1 ) ) + I ε ⁢ ( u ε ( 2 ) ) + Q ε ⁢ ( u ε ( 2 ) ) + o ⁢ ( 1 ) .

Indeed,

| I ε ⁢ ( u ε ) - I ε ⁢ ( u ε ( 1 ) ) - I ε ⁢ ( u ε ( 2 ) ) | ≤ | ∫ ℝ N ( - Δ ) s 2 ⁢ u ε ( 1 ) ⁢ ( - Δ ) s 2 ⁢ u ε ( 2 ) ⁢ 𝑑 x | + ∫ A ε | V ⁢ ( ε ⁢ x ) ⁢ u ε ( 1 ) ⁢ u ε ( 2 ) | ⁢ 𝑑 x

+ ∫ A ε | F ⁢ ( u ε ) - F ⁢ ( u ε ( 1 ) ) - F ⁢ ( u ε ( 2 ) ) | ⁢ 𝑑 x

= : ( I ) + ( II ) + ( III ) .

The second piece can easily be estimated by the boundedness of φε and V, and the information on the 2-norm given by (4.14), i.e., (II)≤Cnε. Similarly, as regards (III), we estimate each single piece separately, in the same way: use (2.5) and the information on the 2-norm and the (p+1)-norm given by (4.14), obtaining (III)≤Cnε.

Focus instead on (I). Recall that (uε)ε⊂S⁢(r2′), and thus ∥uε∥2 is bounded. We have

= : 2 C ( ( I 1 ) + ( I 2 ) )

since on Ωnε⁢iεε×Ωnε⁢iεε and ∁⁢(Ωnε⁢(iε+1)ε)×∁⁢(Ωnε⁢(iε+1)ε) the integrand is null. Focusing on (I1),

( I 1 ) = ∫ ( Ω n ε ⁢ i ε ε × ∁ ( Ω n ε ⁢ ( i ε + 1 ) ε ) ) ∩ { | x - y | > n ε } | u ε ⁢ ( x ) ⁢ u ε ⁢ ( y ) | | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

≤ 1 2 ⁢ ∫ ( Ω n ε ⁢ i ε ε × ∁ ( Ω n ε ⁢ ( i ε + 1 ) ε ) ) ∩ { | x - y | > n ε } u ε 2 ⁢ ( x ) + u ε 2 ⁢ ( y ) | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

= 1 2 ⁢ ∫ Ω n ε ⁢ i ε ε u ε 2 ⁢ ( x ) ⁢ ∫ ∁ ( Ω n ε ⁢ ( i ε + 1 ) ε ) ∩ { | x - y | > n ε } 1 | x - y | N + 2 ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x

+ 1 2 ⁢ ∫ ∁ ⁢ ( Ω n ε ⁢ ( i ε + 1 ) ε ) u ε 2 ⁢ ( y ) ⁢ ∫ Ω n ε ⁢ i ε ε ∩ { | x - y | > n ε } 1 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

≤ C ⁢ ∥ u ε ∥ 2 2 ⁢ ∫ | x - y | > n ε 1 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ C n ε 2 ⁢ s → 0 as ε → 0 .

Focusing on (I2), we have

( I 2 ) ≤ ∫ A ε × ℝ N 1 | x - y | N + 2 ⁢ s ⁢ ( | ( φ ε ⁢ ( x ) - φ ε ⁢ ( y ) ) ⁢ u ε ⁢ ( x ) | + | φ ε ⁢ ( y ) ⁢ ( u ε ⁢ ( x ) - u ε ⁢ ( y ) ) | )

⋅ ( | ( φ ε ( y ) - φ ε ( x ) ) u ε ( x ) | + | ( 1 - φ ε ( y ) ) ( u ε ( x ) - u ε ( y ) ) | ) d x d y

≤ ∫ A ε × ℝ N | φ ε ⁢ ( x ) - φ ε ⁢ ( y ) | 2 ⁢ | u ε ⁢ ( x ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ A ε × ℝ N | u ε ⁢ ( x ) - u ε ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

+ 2 ⁢ ∫ A ε × ℝ N | φ ε ⁢ ( x ) - φ ε ⁢ ( y ) | ⁢ | u ε ⁢ ( x ) | ⁢ | u ε ⁢ ( x ) - u ε ⁢ ( y ) | | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

+ 2 ⁢ ( ∫ A ε × ℝ N | φ ε ⁢ ( x ) - φ ε ⁢ ( y ) | 2 ⁢ | u ε ⁢ ( x ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) 1 2 ⁢ ( ∫ A ε × ℝ N | u ε ⁢ ( x ) - u ε ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) 1 2

= : A 2 + B 2 + 2 A B

and we see that both A and B go to zero: B=[uε]Aε,ℝN≤Cnε1/2 by (4.14), while for A we exploit that ∥∇⁡φε∥∞→0. Indeed, let αε:=1∥∇⁡φε∥∞; we have

A 2 = ∫ A ε | u ε ⁢ ( x ) | 2 ⁢ ∫ | x - y | ≤ α ε | φ ε ⁢ ( x ) - φ ε ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x + ∫ A ε | u ε ⁢ ( x ) | 2 ⁢ ∫ | x - y | > α ε | φ ε ⁢ ( x ) - φ ε ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 y ⁢ 𝑑 x

≤ C ⁢ ∥ u ε ∥ L 2 ⁢ ( A ε ) 2 ⁢ ( ∥ ∇ ⁡ φ ε ∥ ∞ 2 ⁢ ∫ | z | ≤ α ε 1 | z | N + 2 ⁢ s - 2 ⁢ 𝑑 z + 4 ⁢ ∥ φ ε ∥ ∞ 2 ⁢ ∫ | z | > α ε 1 | z | N + 2 ⁢ s ⁢ 𝑑 z )

≤ C n ε ⁢ α ε - 2 ⁢ s ⁢ ( ∥ ∇ ⁡ φ ε ∥ ∞ 2 ⁢ α ε 2 + 1 ) = C n ε ⁢ ∥ ∇ ⁡ φ ε ∥ ∞ 2 ⁢ s ≤ C n ε 2 ⁢ s + 1 → 0 as ε → 0 .

Thus (I2)≤Cnε2⁢s+1+Cnεs+1+Cnε≤C′nε→0, which reaches the claim.

Step 4. Relations of the Derivatives. We have

(4.17) ∥ I ε ′ ⁢ ( u ε ) - I ε ′ ⁢ ( u ε ( 1 ) ) - I ε ′ ⁢ ( u ε ( 2 ) ) ∥ H - s ⁢ ( ℝ N ) → 0 as ε → 0 ,

from which, joined to (4.15),

(4.18) J ε ′ ⁢ ( u ε ) = I ε ′ ⁢ ( u ε ( 1 ) ) + I ε ′ ⁢ ( u ε ( 2 ) ) + Q ε ′ ⁢ ( u ε ( 2 ) ) + o ⁢ ( 1 ) .

Indeed, by Hölder inequality, for any v∈Hs⁢(ℝN),

| I ε ′ ⁢ ( u ε ) ⁢ v - I ε ′ ⁢ ( u ε ( 1 ) ) ⁢ v - I ε ′ ⁢ ( u ε ( 2 ) ) ⁢ v | ≤ ∫ A ε | f ⁢ ( u ε ) - f ⁢ ( u ε ( 1 ) ) - f ⁢ ( u ε ( 2 ) ) | ⁢ | v | ⁢ 𝑑 x

and again we argue in the same way as in the third piece of Step 3, observing that, by (2.5),

| f ⁢ ( u ε ) | ⁢ | v | ≤ β ⁢ | u ε | ⁢ | v | + C β ⁢ | u ε | p ⁢ | v | ,

thus

∫ A ε | f ⁢ ( u ε ) | ⁢ | v | ⁢ 𝑑 x ≤ β ⁢ ∥ u ε ∥ L 2 ⁢ ( A ε ) ⁢ ∥ v ∥ 2 + C β ⁢ ∥ u ε ∥ L p + 1 ⁢ ( A ε ) p ⁢ ∥ v ∥ p + 1 ≤ C ⁢ ( β ⁢ ∥ u ε ∥ L 2 ⁢ ( A ε ) + C β ⁢ ∥ u ε ∥ L p + 1 ⁢ ( A ε ) p ) ⁢ ∥ v ∥ H s

and hence the claim. In particular, |(Iε′⁢(uε)-Iε′⁢(uε(1))-Iε′⁢(uε(2)))⁢uε(2)|≤Cnε. We see also that

(4.19) I ε ′ ⁢ ( u ε ( 1 ) ) ⁢ u ε ( 2 ) = o ⁢ ( 1 ) .

Indeed,

where for (I) and (II) we argue as in Step 3 obtaining (I)+(II)≤Cnε2⁢s+Cnε, while for (III) we argue as in (4.18) obtaining (III)≤Cnε.

Step 5. Convergence of uε(2). Observing that the support of uε(2) is outside Ω/ε, we have with arguments similar to Step 3 that, by (4.13),

(4.20) ∥ u ε ( 2 ) ∥ H s ⁢ ( ℝ N ) ≤ r 1 .

Indeed, focusing only on the nonlocal part, we have (recall that supp(uε(2))⊂∁⁢(Ω/ε))

∫ ℝ 2 ⁢ N | u ε ( 2 ) ⁢ ( x ) - u ε ( 2 ) ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ≤ 2 ⁢ ∫ ∁ ⁢ ( Ω / ε ) × ℝ N | u ε ( 2 ) ⁢ ( x ) - u ε ( 2 ) ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

≤ 4 ⁢ ∫ ∁ ⁢ ( Ω / ε ) × ℝ N | φ ε ⁢ ( x ) - φ ε ⁢ ( y ) | 2 ⁢ | u ε ⁢ ( x ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + 4 ⁢ ∫ ∁ ⁢ ( Ω / ε ) × ℝ N | u ε ⁢ ( x ) - u ε ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

and we use again the final argument in Step 3 and (4.13) to gain, for ε small,

∥ u ε ( 2 ) ∥ H s ⁢ ( ℝ N ) 2 ≤ ( C + o ⁢ ( 1 ) ) ⁢ ∥ u ε ∥ L 2 ⁢ ( ∁ ⁢ ( Ω / ε ) ) 2 + C ⁢ [ u ε ] ∁ ⁢ ( Ω / ε ) , ℝ N 2 ≤ ( C ⁢ r 2 ′ ) 2 ,

where C does not depend on r2′. We choose thus r2′ such that (4.20) holds.

This allows us to use Lemma 4.2. By joining (4.11), (4.18), (4.19), (4.4), (4.7) we obtain

o ⁢ ( 1 ) = J ε ′ ⁢ ( u ε ) ⁢ u ε ( 2 ) = I ε ′ ⁢ ( u ε ( 1 ) ) ⁢ u ε ( 2 ) + I ε ′ ⁢ ( u ε ( 2 ) ) ⁢ u ε ( 2 ) + Q ε ′ ⁢ ( u ε ( 2 ) ) ⁢ u ε ( 2 ) + o ⁢ ( 1 )

≥ C ⁢ ∥ u ε ( 2 ) ∥ H s ⁢ ( ℝ N ) 2 + ( p + 1 ) ⁢ Q ε ⁢ ( u ε ( 2 ) ) + o ⁢ ( 1 )

or more precisely (we highlight this for a later use), for some C=C⁢(r2′),

(4.21) o ⁢ ( 1 ) = J ε ′ ⁢ ( u ε ) ⁢ u ε ( 2 ) ≥ C ⁢ ∥ u ε ( 2 ) ∥ H s ⁢ ( ℝ N ) 2 + ( p + 1 ) ⁢ Q ε ⁢ ( u ε ( 2 ) ) - ( C n ε + C n ε 2 ⁢ s ) ,

which implies (since Qε is positive) that

(4.22) ∥ u ε ( 2 ) ∥ H s ⁢ ( ℝ N ) → 0 as ε → 0

and

(4.23) Q ε ⁢ ( u ε ( 2 ) ) → 0 as ε → 0 .

As a further consequence, (4.22) and the boundedness of V imply

(4.24) I ε ⁢ ( u ε ( 2 ) ) → 0 and I ε ′ ⁢ ( u ε ( 2 ) ) → 0 as ε → 0 .

Step 6. Convergence of Iε′⁢(uε(1)). In particular, we obtain from (4.24), together with (4.16) and (4.23), that

(4.25) J ε ⁢ ( u ε ) = I ε ⁢ ( u ε ( 1 ) ) + o ⁢ ( 1 ) .

We want now to show that

(4.26) I ε ′ ⁢ ( u ε ( 1 ) ) → 0 in H - s ⁢ ( ℝ N ) as ε → 0 .

Start observing that (4.24) together with (4.17) give

(4.27) I ε ′ ⁢ ( u ε ) = I ε ′ ⁢ ( u ε ( 1 ) ) + o ⁢ ( 1 ) ;

let now v∈Hs⁢(ℝN) and evaluate Iε′⁢(uε(1))⁢v. We want to exploit (4.27) together again with assumption (4.11). In order to do this we need to pass from uε(2) to uε, but getting rid of Qε′⁢(uε) on which we have no information. Thus we introduce a cutoff function φ~∈Cc∞⁢(ℝN) such that

Ω 3 2 ⁢ h 0 ≺ φ ~ ≺ Ω 2 ⁢ h 0

and hence

supp ( u ε ( 1 ) ) ⊂ Ω h 0 / ε ⊂ Ω 3 2 ⁢ h 0 / ε ⊂ { φ ~ ( ε ⋅ ) ≡ 1 } ⊂ supp ( φ ~ ( ε ⋅ ) ) ⊂ Ω 2 ⁢ h 0 / ε .

Thus we have

I ε ′ ( u ε ( 1 ) ) v = ( * ) I ε ′ ( u ε ( 1 ) ) ( φ ~ ( ε ⋅ ) v ) + ( 1 + ∥ v ∥ 2 ) o ( 1 )

= I ε ′ ( u ε ) ( φ ~ ( ε ⋅ ) v ) - ( I ε ′ ( u ε ) ( φ ~ ( ε ⋅ ) v ) - I ε ′ ( u ε ( 1 ) ) ( φ ~ ( ε ⋅ ) v ) ) + ( 1 + ∥ v ∥ 2 ) o ( 1 )

= J ε ′ ( u ε ) ( φ ~ ( ε ⋅ ) v ) - ( I ε ′ ( u ε ) - I ε ′ ( u ε ( 1 ) ) ) ( φ ~ ( ε ⋅ ) v ) + ( 1 + ∥ v ∥ 2 ) o ( 1 ) .

Indeed, we justify (*) as done in Step 3: notice that uε(1)≡0 outside Ωh0/ε and 1-φ~(ε⋅)≡0 in Ω32⁢h0/ε, so in the annulus (Ω32⁢h0/ε)∖(Ωh0/ε) both uε(1) and 1-φ~(ε⋅) are zero; notice also that ∁⁢(Ω32⁢h0/ε) and Ωh0/ε get far one from the other as ε→0. Thus we have

| I ε ′ ( u ε ( 1 ) ) ( ( 1 - φ ~ ( ε ⋅ ) ) v ) | = 2 ∫ ( Ω h 0 / ε ) × ∁ ⁢ ( Ω 3 2 ⁢ h 0 / ε ) | u ε ( 1 ) ⁢ ( x ) | ⁢ | ( 1 - φ ~ ⁢ ( ε ⁢ y ) ) ⁢ v ⁢ ( y ) | | x - y | N + 2 ⁢ s d x d y

≤ 2 ⁢ ∫ ( Ω h 0 / ε ) × ∁ ⁢ ( Ω 3 2 ⁢ h 0 / ε ) | u ε ⁢ ( x ) | ⁢ | v ⁢ ( y ) | | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ≤ ( 1 + ∥ v ∥ 2 ) ⁢ o ⁢ ( 1 ) ,

where in the last passage we argue as for (I1) in Step 3. Notice that o⁢(1) does not depend on v. Thus we obtain

| I ε ′ ( u ε ( 1 ) ) v | ≤ ( ∥ J ε ′ ( u ε ) ∥ H - s ⁢ ( ℝ N ) + ∥ I ε ′ ( u ε ) - I ε ′ ( u ε ( 1 ) ) ∥ H - s ⁢ ( ℝ N ) ) ∥ φ ~ ( ε ⋅ ) v ∥ H s ⁢ ( ℝ N ) + ( 1 + ∥ v ∥ H s ⁢ ( ℝ N ) ) o ( 1 )

≤ ( ∥ J ε ′ ⁢ ( u ε ) ∥ H - s ⁢ ( ℝ N ) + ∥ I ε ′ ⁢ ( u ε ) - I ε ′ ⁢ ( u ε ( 1 ) ) ∥ H - s ⁢ ( ℝ N ) ) ⁢ ∥ v ∥ H s ⁢ ( ℝ N ) ⁢ ( C + o ⁢ ( 1 ) ) + ( 1 + ∥ v ∥ H s ⁢ ( ℝ N ) ) ⁢ o ⁢ ( 1 ) ,

where in the last inequality we argue as in Step 5 (again o⁢(1) does not depend on v). Concluding, we have, by choosing ∥v∥Hs⁢(ℝN)=1, that

∥ I ε ′ ⁢ ( u ε ( 1 ) ) ∥ H - s ⁢ ( ℝ N ) ≤ C ⁢ ( ∥ J ε ′ ⁢ ( u ε ) ∥ H - s ⁢ ( ℝ N ) + ∥ I ε ′ ⁢ ( u ε ) - I ε ′ ⁢ ( u ε ( 1 ) ) ∥ H - s ⁢ ( ℝ N ) ) ⁢ ( 1 + o ⁢ ( 1 ) ) + o ⁢ ( 1 ) → 0

by using (4.11) and (4.27).

Step 7. Weak Convergence of uε(1). Set qε:=Υ⁢(uε) to avoid cumbersome notation. Since (ε⁢qε)ε⊂Ω⁢[0,ν0]⊂Ω bounded in ℝN, we have that, up to a subsequence,

ε ⁢ q ε → p 0 ∈ Ω ⁢ [ 0 , ν 0 ] ¯ ⊂ K d ⊂ Ω .

Moreover, by estimating the norm of uε(1) with the norm of uε (as done before, in Step 5, for uε(2)), where uε belongs to S⁢(r2′) bounded in Hs⁢(ℝN), we have that also uε(1) is a bounded sequence, and thus is so uε(1)(⋅+qε), which implies, up to a subsequence,

u ε ( 1 ) ( ⋅ + q ε ) ⇀ U ~ in H s ⁢ ( ℝ N ) as ε → 0 .

For each v∈Hs⁢(ℝN) we apply this weak convergence to the following equalities derived from (4.26):

o ( 1 ) = I ε ′ ( u ε ( 1 ) ) v ( ⋅ - q ε )

= ∫ ℝ N ( - Δ ) s 2 ⁢ u ε ( 1 ) ⁢ ( y + q ε ) ⁢ ( - Δ ) s 2 ⁢ v ⁢ ( y ) ⁢ 𝑑 y + ∫ ℝ N V ⁢ ( ε ⁢ y + ε ⁢ q ε ) ⁢ u ε ( 1 ) ⁢ ( y + q ε ) ⁢ v ⁢ ( y ) ⁢ 𝑑 y - ∫ ℝ N f ⁢ ( u ε ( 1 ) ⁢ ( y + q ε ) ) ⁢ v ⁢ ( y ) ⁢ 𝑑 y

= : ( I ) + ( II ) + ( III ) .

For (I) and (III) we obtain by the weak convergence and classical arguments

( I ) → ∫ ℝ N ( - Δ ) s 2 ⁢ U ~ ⁢ ( - Δ ) s 2 ⁢ v ⁢ 𝑑 y and ( III ) → - ∫ ℝ N f ⁢ ( U ~ ) ⁢ v ⁢ 𝑑 y as ε → 0 .

For (II) instead we have

| ( II ) - ∫ ℝ N V ( p 0 ) U ~ v d y | ≤ ( ∫ ℝ N ( V ( ε y + ε q ε ) - V ( p 0 ) ) 2 v 2 ( y ) d y ) 1 2 ∥ u ε ( 1 ) ( ⋅ + q ε ) ∥ 2

+ V ⁢ ( p 0 ) ⁢ | ∫ ℝ N u ε ( 1 ) ⁢ ( y + q ε ) ⁢ v ⁢ ( y ) ⁢ 𝑑 y - ∫ ℝ N U ~ ⁢ v ⁢ 𝑑 y | ,

where the first term goes to zero (thanks to the boundedness of uε(1)) by the dominated convergence theorem, while the second thanks to the weak convergence. Thus we finally obtain

L V ⁢ ( p 0 ) ′ ⁢ ( U ~ ) ⁢ v = ∫ ℝ N ( - Δ ) s 2 ⁢ U ~ ⁢ ( - Δ ) s 2 ⁢ v ⁢ 𝑑 y + ∫ ℝ N V ⁢ ( p 0 ) ⁢ U ~ ⁢ v ⁢ 𝑑 y - ∫ ℝ N f ⁢ ( U ~ ) ⁢ v ⁢ 𝑑 y = 0

for each v∈Hs⁢(ℝN), that is,

(4.28) L V ⁢ ( p 0 ) ′ ⁢ ( U ~ ) = 0 .

Step 8. Strong Convergence of uε(1). We want to show the strong convergence of uε(1)(⋅+qε), that is,

(4.29) u ε ( 1 ) ( ⋅ + q ε ) → U ~ in H s ⁢ ( ℝ N ) as ε → 0 .

Set w~ε:=uε(1)(⋅+qε)-U~⇀0, again by (4.26) we have

o ( 1 ) = I ε ′ ( u ε ( 1 ) ) w ~ ε ( ⋅ - q ε )

= L V ⁢ ( p 0 ) ′ ⁢ ( U ~ ) ⁢ w ~ ε + ( ∥ ( - Δ ) s 2 ⁢ w ~ ε ∥ 2 2 + ∫ ℝ N V ⁢ ( ε ⁢ y + ε ⁢ q ε ) ⁢ w ~ ε 2 ⁢ 𝑑 y )

+ ∫ ℝ N ( V ⁢ ( ε ⁢ y + ε ⁢ q ε ) - V ⁢ ( p 0 ) ) ⁢ U ~ ⁢ w ~ ε ⁢ 𝑑 y + ∫ ℝ N ( f ⁢ ( U ~ ) - f ⁢ ( U ~ + w ~ ε ) ) ⁢ w ~ ε ⁢ 𝑑 y

= : ( ∥ ( - Δ ) s 2 w ~ ε ∥ 2 2 + ∫ ℝ N V ( ε y + ε q ε ) w ~ ε 2 d y ) + ( I )

≥ ∥ ( - Δ ) s 2 ⁢ w ~ ε ∥ 2 2 + V ¯ ⁢ ∥ w ~ ε ∥ 2 2 + ( I ) ,

where we have used (4.28). We obtain by the boundedness of V and (2.5),

( I ) ≥ - 2 ∥ V ∥ ∞ ∫ ℝ N | U ~ | | w ~ ε | d y - ∫ ℝ N ( 2 β | U | ~ + C β ( 2 p + 1 ) | U ~ | p ) | w ~ ε | d y - ∫ ℝ N ( β | w ~ ε | 2 + 2 p C β | w ~ ε | p + 1 ) d y

= o ⁢ ( 1 ) - β ⁢ ∥ w ~ ε ∥ 2 2 - 2 p ⁢ C β ⁢ ∥ w ~ ε ∥ p + 1 p + 1 ;

in the last passage we have used that w~ε⇀0 in Hs⁢(ℝN), thus by classical properties |w~ε|⇀0 in Hs⁢(ℝN) and hence in L2⁢(ℝN) and in Lp+1⁢(ℝN) (observing that U~p∈L1+1p⁢(ℝN)). Merging together all the things we have, by (4.6) and choosing β<12⁢V¯,

o ⁢ ( 1 ) ≥ ∥ ( - Δ ) s 2 ⁢ w ~ ε ∥ 2 2 + ( V ¯ - β ) ⁢ ∥ w ~ ε ∥ 2 2 - 2 p ⁢ C β ⁢ ∥ w ~ ε ∥ p + 1 p + 1 ≥ C ⁢ ∥ w ~ ε ∥ H s ⁢ ( ℝ N ) 2

and thus w~ε→0 strongly in Hs⁢(ℝN), that is, the claim.

Step 9. Localization. Observe first that U~≢0. Indeed, if not, by (4.22), (4.29) and translation invariance of the norm we would have

r * ≤ lim inf ε → 0 ⁡ ∥ U ε ∥ H s ⁢ ( ℝ N )

≤ lim inf ε → 0 ∥ U ε ( ⋅ - p ε ) + φ ε ∥ H s ⁢ ( ℝ N ) + lim inf ε → 0 ∥ φ ε ∥ H s ⁢ ( ℝ N )

≤ lim ε → 0 ⁡ ( ∥ u ε ( 1 ) ∥ H s ⁢ ( ℝ N ) + ∥ u ε ( 2 ) ∥ H s ⁢ ( ℝ N ) ) + r 2 ′ ≤ r 2 ′ < r * ,

impossible. By (4.29) we obtain also

I ε ⁢ ( u ε ( 1 ) ) → L V ⁢ ( p 0 ) ⁢ ( U ~ ) as ε → 0 .

Thus we find, by using also (4.25) and (4.12),

L V ⁢ ( p 0 ) ⁢ ( U ~ ) = I ε ⁢ ( u ε ( 1 ) ) + o ⁢ ( 1 ) = J ε ⁢ ( u ε ) + o ⁢ ( 1 ) ≤ l 0 ′ + o ⁢ ( 1 )

and hence, letting ε→0,

(4.30) L V ⁢ ( p 0 ) ⁢ ( U ~ ) ≤ l 0 ′ < E m 0 + ν 1 .

Moreover, by (4.28) and U~≢0, we have

E V ⁢ ( p 0 ) ≤ L V ⁢ ( p 0 ) ⁢ ( U ~ ) ;

joining together the two previous inequalities we find EV⁢(p0)<Em0+ν1 which implies, by the monotonicity of Ea, that

V ⁢ ( p 0 ) < m 0 + ν 1 .

Joining this information to the fact that p0∈Ω (and in particular V⁢(p0)≥m0), we have p0∈Ω⁢[0,ν1), that is,

(4.31) ε ⁢ Υ ⁢ ( u ε ) → p 0 ∈ Ω ⁢ [ 0 , ν 1 ) as ε → 0 .

Exploiting again (4.30) (observe that Em0+ν1<Em0+ν0<l0) together with LV⁢(p0)′⁢(U~)=0, and U~≠0, we have that U~ belongs to SV⁢(p0) up to translations, that is,

U := U ~ ( ⋅ - y 0 ) ∈ S V ⁢ ( p 0 ) ⊂ S ^

for some suitable y0∈ℝN. So, set

p ε := q ε + y 0

we have

(4.32) ∥ u ε ( 1 ) - U ( ⋅ - p ε ) ∥ H s ⁢ ( ℝ N ) → 0 as ε → 0 .

For a later use observe also that

(4.33) ε ⁢ p ε → p 0 ∈ Ω ⁢ [ 0 , ν 1 ) as ε → 0 .

Step 10. Conclusions. By (4.31) we have that

ε ⁢ Υ ⁢ ( u ε ) ∈ Ω ⁢ [ 0 , ν 1 )

definitely for ε small. This is the first part of the claim. Moreover, by (4.32) and (4.22) we have

(4.34) ∥ u ε - U ( ⋅ - p ε ) ∥ H s ⁢ ( ℝ N ) → 0 as ε → 0

and thus, since ρ^(uε)≤∥uε-U(⋅-pε)∥Hs⁢(ℝN) by definition, also ρ^⁢(uε)→0 and hence

ρ ^ ⁢ ( u ε ) ∈ [ 0 , ρ 1 ]

definitely for ε small. This concludes the proof. ∎

In the next proposition we see that solutions of Jε′⁢(u)=0 are, under suitable assumptions, also solutions of Iε′⁢(u)=0.

Corollary 4.9.

Let (uε)ε be a sequence of critical points of Jε, that is, Jε′⁢(uε)=0, satisfying

u ε ∈ S ⁢ ( r 2 ′ ) , J ε ⁢ ( u ε ) ≤ l 0 ′ 𝑎𝑛𝑑 ε ⁢ Υ ⁢ ( u ε ) ∈ Ω ⁢ [ 0 , ν 0 ]

for any ε>0. Then, for ε sufficiently small, we have

Q ε ⁢ ( u ε ) = 0 , 𝑎𝑛𝑑 Q ε ′ ⁢ ( u ε ) = 0 .

In particular, Iε′⁢(uε)=0, which means that uε is a solution of (4.1).

Proof.

By the proof of Lemma 4.6, we notice, since 1-φiεε≡1 outside Ωiε+1ε, and thus outside Ωh0/ε, that

(4.35) ∥ u ε ∥ L 2 ⁢ ( ℝ N ∖ ( Ω h 0 / ε ) ) = ∥ u ε ( 2 ) ∥ L 2 ⁢ ( ℝ N ∖ ( Ω h 0 / ε ) ) ≤ ∥ u ε ( 2 ) ∥ H s ⁢ ( ℝ N )

and hence ∥uε∥L2⁢(ℝN∖(Ωh0/ε))→0 by (4.22).

Through a careful analysis of Steps 3–5 of the proof, that is, by (4.21) and (4.35), we see, more precisely, that

∥ u ε ∥ L 2 ⁢ ( ℝ N ∖ ( Ω h 0 / ε ) ) 2 ≤ C n ε + C n ε 2 ⁢ s + o ⁢ ( 1 )

where C=C⁢(r2′) and o⁢(1) depends on the rate of convergence of Jε′⁢(uε). Thus, since we assume Jε′⁢(uε)≡0, we gain uniformity, i.e., called α*:=min⁡{1,2⁢s}, we obtain

∥ u ε ∥ L 2 ⁢ ( ℝ N ∖ ( Ω h 0 / ε ) ) 2 ≤ C n ε α * ∼ ε α * 2

As a consequence

1 ε α ⁢ ∥ u ε ∥ L 2 ⁢ ( ℝ N ∖ ( Ω h 0 / ε ) ) 2 → 0 as ε → 0

for α∈(0,α*2), and hence Qε⁢(uε)≡Qε′⁢(uε)≡0 for ε sufficiently small. ∎

We want to show now a (truncated) Palais–Smale-like condition.

Proposition 4.10.

There exists r2′′∈(0,min⁡{r0,r1}) sufficiently small with the following property: let ε>0 fixed and let (uj)j⊂S⁢(r2′′) be such that

∥ J ε ′ ⁢ ( u j ) ∥ H - s ⁢ ( ℝ N ) → 0 as j → + ∞

with the additional assumption

( ε ⁢ Υ ⁢ ( u j ) ) j ⊂ Ω ⁢ [ 0 , ν 0 ] .

Then (uj)j admits a strongly convergent subsequence in Hs⁢(RN).

Proof.

Let r2′′ to be fixed. Since S⁢(r2′′) is bounded, up to a subsequence we can assume uj⇀u0 in Hs⁢(ℝN). We want to show that

lim R → + ∞ ⁡ lim j → + ∞ ⁡ ∥ u j ∥ L q ⁢ ( ℝ N ∖ B R ) = 0

for q=2 and q=p+1 and conclude by Lemma 4.4.

Arguing similarly to Step 1 of the proof of Lemma 4.6, i.e., exploiting Remark 4.8, we obtain for L≫0, uniformly in j∈ℕ,

|| | u j | || ℝ N ∖ B L ≤ C ⁢ r 2 ′′ ;

indeed, we work with the set BL-Υ⁢(uj) which expands to ℝN as L,j→+∞, since Υ⁢(uj)∈Ω/ε, a fixed bounded set. Moreover, for any n∈ℕ, we have

∑ i = 1 n ∥ u j ∥ L 2 ⁢ ( B L + n ⁢ i ∖ B L + n ⁢ ( i - 1 ) ) 2 ≤ ( C ⁢ r 2 ′′ ) 2

and similarly for the Gagliardo seminorm and the (p+1)-norm, thus for some ij,n∈{1,…,n},

∥ u j ∥ A j , n 2 + ∥ u j ∥ L p + 1 ⁢ ( A j , n ) p + 1 ≤ C n ,

where Aj,n:=BL+n⁢ij,n∖BL+n⁢(ij,n-1). Again similarly to Step 2 of the proof of Lemma 4.6, we introduce ψj,n such that

B L + n ⁢ ( i j , n - 1 ) ≺ ψ j , n ≺ B L + n ⁢ i j , n

and ∥∇⁡ψj,n∥∞=o⁢(1) as n→+∞; moreover, we set

u ~ j , n := ( 1 - ψ j , n ) ⁢ u j .

Observe that χBL≤ψj,n, thus supp(u~j,n)⊂∁⁢(BL). Arguing as in Step 5 and 3 of the proof of Lemma 4.6, we obtain

∫ ℝ 2 ⁢ N | u ~ j , n ⁢ ( x ) - u ~ j , n ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ≤ 4 ⁢ ∫ ∁ ⁢ ( B L ) × ℝ N | ψ j , n ⁢ ( x ) - ψ j , n ⁢ ( y ) | 2 ⁢ | u j ⁢ ( x ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + 4 ⁢ ∫ ∁ ⁢ ( B L ) × ℝ N | u j ⁢ ( x ) - u j ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

≤ o ⁢ ( 1 ) ⁢ ∥ u j ∥ L 2 ⁢ ( ∁ ⁢ ( B L ) ) 2 + C ⁢ [ u j ] ∁ ⁢ ( B L ) , ℝ N 2 ,

thus ∥u~j,n∥Hs⁢(ℝN)≤C⁢r2′′ and hence, choosing r2′′ sufficiently small, we have

∥ u ~ j , n ∥ H s ⁢ ( ℝ N ) ≤ r 1 ;

by Lemma 4.2, for q∈{2,p+1}, we obtain

∥ u j ∥ L q ⁢ ( ℝ N ∖ B L + n ⁢ i j , n ) 2 = ∥ u ~ j , n ∥ L q ⁢ ( ℝ N ∖ B L + n ⁢ i j , n ) 2 ≤ C ⁢ ∥ u ~ j , n ∥ H s ⁢ ( ℝ N ) 2 ≤ C ⁢ I ε ′ ⁢ ( u ~ j , n ) ⁢ u ~ j , n .

Thus the claim comes if we show that

I ε ′ ⁢ ( u ~ j , n ) ⁢ u ~ j , n → 0 as j , n → + ∞ .

Indeed, we have

I ε ′ ⁢ ( u ~ j , n ) ⁢ u ~ j , n = I ε ′ ⁢ ( u j ) ⁢ u ~ j , n - ∫ ℝ 2 ⁢ N ( - Δ ) s 2 ⁢ ( ψ j , n ⁢ u j ) ⁢ ( - Δ ) s 2 ⁢ ( ( 1 - ψ j , n ) ⁢ u j ) ⁢ 𝑑 x

- ∫ A j , n V ⁢ ( ε ⁢ x ) ⁢ ψ j , n ⁢ ( 1 - ψ j , n ) ⁢ u j 2 ⁢ 𝑑 x - ∫ A j , n ( f ⁢ ( ( 1 - ψ j , n ) ⁢ u j ) - f ⁢ ( u j ) ) ⁢ ( 1 - ψ j , n ) ⁢ u j ⁢ 𝑑 x

= : J ε ′ ( u j ) u ~ j , n - Q ε ′ ( u j ) u ~ j , n + ( I ) ≤ o ( 1 ) + ( I ) ,

where we have used that Jε′⁢(uj)→0 (as j→+∞, uniformly in n∈ℕ), the boundedness of ∥u~j,n∥Hs⁢(ℝN) and the positivity of Qε′⁢(uj)⁢u~j,n. The term (I) can be estimated in the same way as done in Steps 3–4 of the proof of Lemma 4.6 (fixed j∈ℕ, and n→+∞), and hence we reach the claim. ∎

4.3 Deformation Lemma on a Neighborhood of Expected Solutions

We want to define now a neighborhood of expected solutions (see [21]), which will be invariant under a suitable deformation flow. Consider r3:=min⁡{r′,r0′,r2′,r2′′} (see Lemma 3.5, Lemma 4.1, Theorem 4.7 and Proposition 4.10), and let us define

R ⁢ ( δ , u ) := δ - δ 2 2 ⁢ ( ρ ^ ⁢ ( u ) - ρ 1 ) + ≤ δ

and

𝒳 ε , δ := { u ∈ S ⁢ ( ρ 0 ) ∣ ε ⁢ Υ ⁢ ( u ) ∈ Ω ⁢ [ 0 , ν 0 ) , J ε ⁢ ( u ) < E m 0 + R ⁢ ( δ , u ) } ,

where

0 < ρ 1 < ρ 0 < r 3 ,

ε is sufficiently small and

(4.36) δ ∈ ( 0 , min ⁡ { δ 2 4 ⁢ ( ρ 0 - ρ 1 ) , δ 1 , l 0 ′ - E m 0 } ) ;

here δ1 and δ2 are the ones that appear in Lemma 4.1 and Theorem 4.7. Notice that the height of the sublevel in 𝒳ε,δ depends on u itself; this will be used to gain a deformation which preserves 𝒳ε,δ.

We begin by pointing out some geometrical features of the neighborhood 𝒳ε,δ.

𝒳 ε , δ is open. Indeed, S⁢(ρ) and {Jε⁢(u)<Em0+R⁢(δ,u)} are open, and Ω⁢[0,ν0)=Ω⁢(-γ,ν0) for a whatever γ>0 (since V cannot go under m0 in Ω) and thus open. Moreover, it is nonempty (see e.g. Section 4.4.1).
If v∈𝒳ε,δ⊂S⁢(ρ0), then by (3.7) we have ρ^⁢(v)<ρ0.
If v∈𝒳ε,δ⊂{εΥ(v)∈Ω[0,ν0]}, then

(4.37) ε ⁢ Υ ⁢ ( v ) ∈ Ω ⁢ [ 0 , ν 1 ) .

Indeed, if not, i.e., ε⁢Υ⁢(v)∈Ω⁢[ν1,ν0], then by Lemma 4.1 we have

J ε ⁢ ( v ) ≥ I ε ⁢ ( v ) ≥ E m 0 + δ 1 > E m 0 + δ ≥ E m 0 + R ⁢ ( δ , v ) ,

which is an absurd.
If R⁢(δ,v)≥-δ, then

(4.38) v ∈ S ⁢ ( ρ 0 ) .

Indeed, δ24⁢(ρ^⁢(u)-ρ1)+≤δ implies, by the restriction on δ,

( ρ ^ ⁢ ( u ) - ρ 1 ) + < ρ 0 - ρ 1 .

If ρ^⁢(u)<ρ1, clearly u∈S⁢(ρ1)⊂S⁢(ρ0). If instead ρ^⁢(u)≥ρ1, then ρ^⁢(u)<ρ0, which again implies u∈S⁢(ρ0).

We further define the set of critical points of Jε lying in the neighborhood of expected solutions

K c := { u ∈ 𝒳 ε , δ ∣ J ε ′ ⁢ ( u ) = 0 , J ε ⁢ ( u ) = c } ,

the sublevel

𝒳 ε , δ c := 𝒳 ε , δ ∩ J ε c

and

( 𝒳 ε , δ ) d c := { u ∈ 𝒳 ε , δ ∣ d ≤ J ε ⁢ ( u ) ≤ c }

for every c,d∈ℝ. We present now a deformation lemma with respect to Kc, for c sufficiently close to Em0.

Lemma 4.11.

Let c∈(Em0-δ,Em0+δ). Then there exists a deformation at level c, which leaves the set Xε,δ invariant. That is, for every U neighborhood of Kc (U=∅ if Kc=∅), there exist a small ω>0 and a continuous deformation η:[0,1]×Xε,δ→Xε,δ such that

η ⁢ ( 0 , ⋅ ) = id ,
J ε ⁢ ( η ⁢ ( ⋅ , u ) ) is non-increasing,
η ⁢ ( t , u ) = u for every t ∈ [ 0 , 1 ] , if J ε ⁢ ( u ) ∉ ( E m 0 - δ , E m 0 + δ ) ,
η ⁢ ( 1 , 𝒳 ε , δ c + ω ∖ U ) ⊂ 𝒳 ε , δ c - ω ,
η ⁢ ( ⋅ , u ) is a semigroup.

Proof.

Let 𝒱:{u∈Hs⁢(ℝN)∣Jε′⁢(u)≠0}→Hs⁢(ℝN) be a locally Lipschitz pseudo-gradient vector field associated to Jε, and let ϕ∈Liploc⁢(Hs⁢(ℝN),ℝ) be a cutoff function such that supp(ϕ)⊂(𝒳ε,δ)Em0-δEm0+δ and ϕ=1 in a small neighborhood of c. We consider the Cauchy problem

(4.39) { η ˙ = - ϕ ⁢ ( η ) ⁢ 𝒱 ⁢ ( η ) ∥ 𝒱 ⁢ ( η ) ∥ H s ⁢ ( ℝ N ) , η ⁢ ( 0 , u ) = u .

The proof keeps on classically, obtaining a deformation η:[0,1]×𝒳ε,δ→Hs⁢(ℝN). We want to prove now that η goes into 𝒳ε,δ.

Let u∈𝒳ε,δ. We need to show that η⁢(t,u)∈𝒳ε,δ for every t>0. Since 𝒳ε,δ is open, η⁢(s,u) continues staying in 𝒳ε,δ for s small. Thus assume that

η ⁢ ( s , u ) ∈ 𝒳 ε , δ for every 0 ≤ s < t 0

for some t0>0, and we want to show that η⁢(t0,u)∈𝒳ε,δ. Notice first that, by using (iii), (v), (ii) and the continuity of η and Jε we can assume that

(4.40) J ε ⁢ ( η ⁢ ( t 0 , u ) ) ∈ [ E m 0 - δ , E m 0 + δ ) .

Step 1: ε⁢Υ⁢(η⁢(t0,u))∈Ω⁢[0,ν0). By (4.37) we have

ε ⁢ Υ ⁢ ( η ⁢ ( s , u ) ) ∈ Ω ⁢ [ 0 , ν 1 ) for every 0 ≤ s < t 0 ,

and thus by continuity ε⁢Υ⁢(η⁢(t0,u))∈Ω⁢[0,ν1)¯⊂Ω⁢[0,ν0].

Step 2: Jε⁢(η⁢(t0,u))<Em0+R⁢(δ,η⁢(t0,u)). If ρ^⁢(η⁢(t0,u))≤ρ1, then R⁢(δ,η⁢(t0,u))=δ and we directly have the claim, recalled that Jε⁢(η⁢(t0,u))<Em0+δ by (4.40). Assume instead ρ^⁢(η⁢(t0,u))>ρ1. By continuity, there exists t1∈(0,t0) such that we have ρ^⁢(η⁢(s,u))>ρ1 for every s∈[t1,t0]. In particular,

{ η ⁢ ( s , u ) ∈ S ⁢ ( ρ 0 ) ⊂ S ⁢ ( r 3 ) ⊂ S ⁢ ( r 2 ′ ) , J ε ⁢ ( η ⁢ ( s , u ) ) < E m 0 + δ < E m 0 + ν 1 , ρ ^ ⁢ ( η ⁢ ( s , u ) ) ∈ ( ρ 1 , ρ 0 ] , ε ⁢ Υ ⁢ ( η ⁢ ( s , u ) ) ∈ Ω ⁢ [ 0 , ν 0 ]

for s∈[t1,t0]. Then by Theorem 4.7 we have

∥ J ε ′ ⁢ ( η ⁢ ( s , u ) ) ∥ H - s ⁢ ( ℝ N ) ≥ δ 2 for every s ∈ [ t 1 , t 0 ] .

We can thus compute with standard argument, by using (4.39), the properties of the pseudo-gradient and (3.8),

J ε ⁢ ( η ⁢ ( t 0 , u ) ) ≤ J ε ⁢ ( η ⁢ ( t 1 , u ) ) - δ 2 2 ⁢ ( ρ ^ ⁢ ( η ⁢ ( t 0 , u ) ) - ρ ^ ⁢ ( η ⁢ ( t 1 , u ) ) )

< E m 0 + δ - δ 2 2 ⁢ ( ρ ^ ⁢ ( η ⁢ ( t 1 , u ) ) - ρ 1 ) - δ 2 2 ⁢ ( ρ ^ ⁢ ( η ⁢ ( t 0 , u ) ) - ρ ^ ⁢ ( η ⁢ ( t 1 , u ) ) )

= E m 0 + R ⁢ ( δ , η ⁢ ( t 0 , u ) ) ,

that is, the claim.

Step 3: η⁢(t0,u)∈S⁢(ρ0). By the previous point we have

J ε ⁢ ( η ⁢ ( t 0 , u ) ) ≤ E m 0 + R ⁢ ( δ , η ⁢ ( t 0 , u ) ) .

Since (4.40) implies Jε⁢(η⁢(t0,u))≥Em0-δ, it follows from (4.38) that η⁢(t0,u)∈S⁢(ρ0), and thus the claim. ∎

4.4 Maps Homotopic to the Embedding

We search now for two maps Φε, Ψε such that, for a sufficiently small σ0∈(0,1) and a sufficiently small δ^=δ^⁢(σ0)∈(0,δ) (see (4.36)), defined

I := [ 1 - σ 0 , 1 + σ 0 ] ,

we have, for small ε,

I × K → Φ ε 𝒳 ε , δ E m 0 + δ ^ → Ψ ε I × K d

with the additional condition

∂ ⁡ I × K → Φ ε 𝒳 ε , δ E m 0 - δ ^ → Ψ ε ( I ∖ { 1 } ) × K d ;

then we will prove that Ψε∘Φε is homotopic to the identity. While the first property is useful for category arguments to gain multiplicity of solutions, the second additional condition will be essential for developing relative category (and cup-length) arguments.

4.4.1 Definition of Φε

Let us fix a ground state U0∈Sm0⊂S^, i.e., Lm0⁢(U0)=Em0 (see Theorem 3.1 and (3.2)). Define, for p∈K and t∈I (σ0 to be fixed)

Φ ε ⁢ ( t , p ) := U 0 ⁢ ( ⋅ - p ε t ) ∈ H s ⁢ ( ℝ N ) .

We show now that, for ε small, Φε⁢(t,p)∈𝒳ε,δEm0+δ^.

Φ ε ⁢ ( t , p ) ∈ S ⁢ ( ρ 1 ) ⊂ S ⁢ ( ρ 0 ) : indeed, recalled that the dilation t∈ℝ↦U0⁢(⋅t)∈Hs⁢(ℝN) is continuous, we have

∥ U 0 ( ⋅ - p ε t ) - U 0 ( ⋅ - p ε ) ∥ H s ⁢ ( ℝ N ) = ∥ U 0 ( ⋅ t ) - U 0 ∥ H s ⁢ ( ℝ N ) < ρ 1

for t∈I and sufficiently small σ0=σ0⁢(U0) (not depending on ε). Thus, setting

φ t := U 0 ( ⋅ - p ε t ) - U 0 ( ⋅ - p ε ) ,

we have

Φ ε ( t , p ) = U 0 ( ⋅ - p ε ) + φ t

with U0∈S^, pε∈ℝN and ∥φt∥Hs⁢(ℝN)<ρ1, which is the claim.
ε ⁢ Υ ⁢ ( Φ ε ⁢ ( t , p ) ) ∈ Ω ⁢ [ 0 , ν 0 ) : indeed, by the previous point and Lemma 3.7, we have

| Υ ⁢ ( Φ ε ⁢ ( t , p ) ) - p ε | < 2 ⁢ R 0 ,

hence |ε⁢Υ⁢(Φε⁢(t,p))-p|<2⁢ε⁢R0, and since p∈K,

d ⁢ ( ε ⁢ Υ ⁢ ( Φ ε ⁢ ( t , p ) ) , K ) < 2 ⁢ ε ⁢ R 0 .

For sufficiently small ε, we have K2⁢ε⁢R0⊂Ω⁢[0,ν0), and thus the claim. In particular, for a later use observe that

(4.41) ε ⁢ Υ ⁢ ( Φ ε ⁢ ( t , p ) ) = p + o ⁢ ( 1 ) .
J ε ⁢ ( Φ ε ⁢ ( t , p ) ) < E m 0 + R ⁢ ( δ , Φ ε ⁢ ( t , p ) ) : indeed, Φε⁢(t,p)∈S⁢(ρ1), thus ρ^⁢(Φε⁢(t,p))<ρ1, which implies

R ⁢ ( δ , Φ ε ⁢ ( t , p ) ) = δ

and the claim comes from the following point, since δ^<δ.
J ε ⁢ ( Φ ε ⁢ ( t , p ) ) < E m 0 + δ ^ : indeed, we have by Lemma 3.3 (b)

J ε ⁢ ( Φ ε ⁢ ( t , p ) ) = L m 0 ⁢ ( Φ ε ⁢ ( t , p ) ) + 1 2 ⁢ ∫ ℝ N ( V ⁢ ( ε ⁢ x ) - m 0 ) ⁢ Φ ε 2 ⁢ ( t , p ) ⁢ 𝑑 x + Q ε ⁢ ( Φ ε ⁢ ( t , p ) )

(4.42) = : L m 0 ( U 0 ( ⋅ - p ε t ) ) + ( I ) + ( II ) = g ( t ) E m 0 + o ( 1 ) ≤ E m 0 + o ( 1 ) ,

where we used g⁢(t)≤1. Indeed, as regards (I) we have

( I ) = 1 2 ⁢ ∫ ℝ N ( V ⁢ ( ε ⁢ x + p ) - m 0 ) ⁢ U 0 2 ⁢ ( x t ) ⁢ 𝑑 x → 0 as ε → 0

by exploiting that p∈K and the dominated convergence theorem, together with the boundedness of V. Focusing on (II) instead, we have

( II ) = ( 1 ε α ⁢ ∥ U 0 ⁢ ( ⋅ t ) ∥ L 2 ( ℝ N ∖ ( ( Ω 2 ⁢ h 0 - p ) / ε ) 2 - 1 ) + p + 1 2 ;

since p∈K⊂Ω⊂Ω2⁢h0, we have 0∈Ω2⁢h0-p and moreover Br⊂Ω2⁢h0-p for some ball Br; notice that Br/ε covers the whole ℝN as ε→0. Therefore, by the polynomial estimate we have

∥ U 0 ⁢ ( ⋅ t ) ∥ L 2 ( ℝ N ∖ ( ( Ω 2 ⁢ h 0 - p ) / ε ) 2 ≤ C ⁢ ∥ 1 1 + | x | N + 2 ⁢ s ∥ L 2 ⁢ ( ℝ N ∖ ( B r / ε ) ) 2 ≤ C ⁢ ε N + 4 ⁢ s ,

and hence (II)→0 as ε→0, since α<N+4⁢s. Therefore, by choosing a sufficiently small ε, we obtain

J ε ⁢ ( Φ ε ⁢ ( t , p ) ) ≤ E m 0 + 1 2 ⁢ δ ^ < E m 0 + δ ^ .

Finally, we show the additional condition.

J ε ⁢ ( Φ ε ⁢ ( 1 ± σ 0 , p ) ) < E m 0 - δ ^ : indeed, looking at (4.42) we see that, for small ε,

J ε ⁢ ( Φ ε ⁢ ( 1 ± σ 0 , p ) ) < g ⁢ ( 1 ± σ 0 ) ⁢ E m 0 + δ ^ ;

since g⁢(1±σ0)<1, we can find a small δ^<1-g⁢(1±σ0)2⁢Em0 (not depending on ε) such that

J ε ⁢ ( Φ ε ⁢ ( 1 ± σ 0 , p ) ) < g ⁢ ( 1 ± σ 0 ) ⁢ E m 0 + δ ^ < E m 0 - δ ^

and thus the claim.

4.4.2 Definition of Ψε

Define a truncation

T ( t ) := { 1 - σ 0 if t ≤ 1 - σ 0 , t if t ∈ ( 1 - σ 0 , 1 + σ 0 ) , 1 + σ 0 if t ≥ 1 + σ 0

for t∈ℝ, and

Ψ ε ⁢ ( u ) := ( T ⁢ ( P m 0 ⁢ ( u ) ) , ε ⁢ Υ ⁢ ( u ) )

for every u∈𝒳ε,δEm0+δ^. By the definition of T and property (4.37), we have directly

Ψ ε ⁢ ( u ) ∈ I × Ω ⁢ [ 0 , ν 1 ] ⊂ I × Ω ⁢ [ 0 , ν 0 ] ⊂ I × K d .

Assume now u∈𝒳ε,δEm0-δ^. We have, by using Lemma 4.5 and Lemma 3.5,

E m 0 - δ ^ ≥ J ε ⁢ ( u ) ≥ L m 0 ⁢ ( u ) - C min ⁢ ε α ≥ g ⁢ ( P m 0 ⁢ ( u ) ) ⁢ E m 0 - C min ⁢ ε α

and hence

E m 0 ≥ g ⁢ ( P m 0 ⁢ ( u ) ) ⁢ E m 0 + δ ^ - C min ⁢ ε α > g ⁢ ( P m 0 ⁢ ( u ) ) ⁢ E m 0 ,

where the last inequality holds for ε small, not depending on u. Thus

g ⁢ ( P m 0 ⁢ ( u ) ) < 1

and this must imply, by the properties of g, that Pm0⁢(u)≠1, and in particular

T ⁢ ( P m 0 ⁢ ( u ) ) ≠ 1 .

This reaches the goal.

4.4.3 An Homotopy to the Identity

Introduce a notation from the algebraic topology: we write, for B⊂A and B′⊂A′,

f : ( A , B ) → ( A ′ , B ′ )

whenever

f ∈ C ⁢ ( A , A ′ ) and f ⁢ ( B ) ⊂ B ′ .

Observed that Φε and Ψε are continuous, we can rewrite the stated properties as

Φ ε : ( I × K , ∂ ⁡ I × K ) → ( 𝒳 ε , δ E m 0 + δ ^ , 𝒳 ε , δ E m 0 - δ ^ ) ,

Ψ ε : ( 𝒳 ε , δ E m 0 + δ ^ , 𝒳 ε , δ E m 0 - δ ^ ) → ( I × K d , ( I ∖ { 1 } ) × K d ) ,

Ψ ε ∘ Φ ε : ( I × K , ∂ ⁡ I × K ) → ( I × K d , ( I ∖ { 1 } ) × K d ) ,

where a straightforward computation shows

( Ψ ε ∘ Φ ε ) ⁢ ( t , p ) = ( t , ε ⁢ Υ ⁢ ( U 0 ⁢ ( ⋅ - p ε t ) ) ) .

Clearly, we notice that the inclusion map has the same property, that is, set j⁢(t,p):=(t,p) we have

j : ( I × K , ∂ ⁡ I × K ) → ( I × K d , ( I ∖ { 1 } ) × K d ) .

We want to show that these maps are homotopic, information useful in the theory of relative cup-length.

Proposition 4.12.

For sufficiently small ε, the maps Ψε∘Φε and j are homotopic, that is, there exists a continuous map H:[0,1]×I×K→I×Kd such that

H ⁢ ( θ , ⋅ , ⋅ ) : ( I × K , ∂ ⁡ I × K ) → ( I × K d , ( I ∖ { 1 } ) × K d )

for each θ∈[0,1], with H⁢(0,⋅,⋅)=Ψε∘Φε and H⁢(1,⋅,⋅)=j.

Proof.

Noticed that also Ψε∘Φε fixes the first variable, it is sufficient to link the second variables through a segment, that is,

H ⁢ ( θ , t , p ) := ( t , ( 1 - θ ) ⁢ ε ⁢ Υ ⁢ ( U 0 ⁢ ( ⋅ - p ε t ) ) + θ ⁢ p ) ,

with θ∈[0,1]. We must check that H is well defined, since Kd is not a convex set, generally. Indeed we have, by (4.41),

( 1 - θ ) ⁢ ε ⁢ Υ ⁢ ( U 0 ⁢ ( ⋅ - p ε t ) ) + θ ⁢ p = ( 1 - θ ) ⁢ p + o ⁢ ( 1 ) + θ ⁢ p = p + o ⁢ ( 1 ) .

Since p∈K, for sufficiently small ε we have that p+o⁢(1)∈Kd, and thus the claim. ∎

5 Existence of Multiple Solutions

We finally come up to the existence of multiple solutions. Here the algebraic notions of relative category and relative cup-length are of key importance.

Remark 5.1.

Before coming up to multiplicity results, we highlight that existence of a single solution could be obtained without any use of algebraic tools. We omit the details, gaining instead the existence as a byproduct of the multiplicity.

In what follows we refer to [48, 36] and references therein, and to [7] for a complete treatment on the topic of category.

Definition 5.2.

Let X be a topological space and let A,B be two closed subsets. We call the category of A in X, relative to B, and write k=catX,B(A), the least integer k∈ℕ such that there exist A0,A1,…,Ak closed subsets of X which verify

( A i ) i = 0 ⁢ … ⁢ k cover A,
( A i ) i = 1 ⁢ … ⁢ k are contractible,
A 0 is deformable in B, i.e., there exists a continuous h0:[0,1]×(A0∪B)→X such that h0⁢(0,⋅)=id, h1⁢(1,A0)⊂B and h0⁢(t,B)⊂B for each t∈[0,1].

If such k does not exists, we set catX,B(A):=+∞. Notice that catX(A)=catX,∅(A).

Since we will work only with X=𝒳ε,δ, to avoid cumbersome notation we will write cat(A):=cat𝒳ε,δ(A) and cat(A,B):=cat𝒳ε,δ,B(A).

We recall the definition of (relative) cup-length (see e.g. [19] and the references therein). Let B⊂A be topological spaces, and let 𝔽 be a whatever field. Let moreover H*⁢(A)=⊕q≥0Hq⁢(A) be the Alexander–Spanier cohomology with coefficients in some field 𝔽 and let H*⁢(A,B) be the relative Alexander–Spanier cohomology (see [41] and the references therein). Let ⌣:H*(A)×H*(A)→H*(A) be the cup product and let ⌣:H*(A,B)×H*(A)→H*(A,B) be the relative cup product. In addition, let us recall that a whatever continuous function f:(A,B)→(A′,B′), being (A′,B′) another topological pair, induces homomorphisms f*:H*⁢(A)→H*⁢(A) and f*:H*⁢(A′,B′)→H*⁢(A,B). We define

cupl ( f ) := max { l ∈ ℕ | there exist α i ∈ H q i ⁢ ( A ′ ) , q i ≥ 1 , for i = 1 ⁢ … ⁢ l , and there exist α 0 ∈ H * ⁢ ( A ′ , B ′ )

such that f * ( α 0 ) ⌣ f * ( α 1 ) ⌣ ⋯ ⌣ f * ( α l ) ≠ 0 in H * ⁢ ( A , B ) } ;

if such l∈ℕ does not exists, we define cupl⁢(f):=-1. Moreover, we set cupl⁢(A,B):=cupl⁢(id(A,B)) and cupl⁢(A):=cupl⁢(A,∅). In the case X is not connected, a slightly different definition (which makes the cup-length additive) can be found in [7]. Here we collect some properties of the cup-length, see [9, Lemma 2.6].

Lemma 5.3.

We have the following properties.

For any f : ( A , B ) → ( A ′ , B ′ ) and f ′ : ( A ′ , B ′ ) → ( A ′′ , B ′′ ) it results that

cupl ⁢ ( f ′ ∘ f ) ≤ min ⁡ { cupl ⁢ ( f ′ ) , cupl ⁢ ( f ) } .

As a consequence,

(5.1) cupl ⁢ ( f ′ ∘ f ) ≤ cupl ⁢ ( A ′ , B ′ ) .
For any f , g : ( A , B ) → ( A ′ , B ′ ) homotopic, we have

cupl ⁢ ( f ) = cupl ⁢ ( g ) .

Finally, we cite the following result ([8]) which can be found in [21, Lemma 5.5].

Lemma 5.4.

Consider the inclusion j:(I×K,∂⁡I×K)→(I×Kd,∂⁡I×Kd) for a whatever K⊂RN compact, Kd:={x∈RN∣d⁢(x,K)≤d}, and I=[a,b]. Then, for d>0 sufficiently small, we have

cupl ⁢ ( j ) ≥ cupl ⁢ ( K ) .

The following lemma links the concepts of category and cup-length, and it can be found in [48, Proposition 2.6] or [36, Theorem 3.6].

Lemma 5.5.

For any B⊂A⊂RN closed, we have

cat ( A , B ) ≥ cupl ⁢ ( A , B ) + 1 .

We are ready now to prove the main theorem.

Proof of Theorem 1.1.

By the construction of the neighborhood 𝒳ε,δ and Corollary 4.9 (recall that ρ0<r3≤r2′ and that Jε⁢(u)<Em0+R⁢(δ^,u)≤Em0+δ^≤l0′ for u∈𝒳ε,δ), we have

{ u ∈ ( 𝒳 ε , δ ) E m 0 - δ ^ E m 0 + δ ^ | J ε ′ ⁢ ( u ) = 0 } ⊂ { u ∈ H s ⁢ ( ℝ N ) ∣ I ε ′ ⁢ ( u ) = 0 } .

Thus we obtain

# ⁢ { u ⁢ solutions of (4.1) } ≥ # ⁢ { u ∈ ( 𝒳 ε , δ ) E m 0 - δ ^ E m 0 + δ ^ | J ε ′ ⁢ ( u ) = 0 }

≥ ( i ) ⁢ cat ( 𝒳 ε , δ E m 0 + δ ^ , 𝒳 ε , δ E m 0 - δ ^ ) ⁢ ≥ ( ii ) ⁢ cupl ⁢ ( 𝒳 ε , δ E m 0 + δ ^ , 𝒳 ε , δ E m 0 - δ ^ ) + 1 ⁢ ≥ ( iii ) ⁢ cupl ⁢ ( K ) + 1 ,

that is, the claim, up to the proof of (i)–(iii). Indeed, inequality (i) is obtained classically from the Deformation Lemma 4.11 (see e.g. [48, Proposition 3.2] or [36, Theorem 6.1]). Inequality (ii) is given by the algebraic-topological Lemma 5.5. Point (iii) is instead due to the existence of the homotopy gained in Proposition 4.12 and properties of the cup-length: indeed, by (5.1) in Lemma 5.3 (a), we have

cupl ⁢ ( 𝒳 ε , δ E m 0 + δ ^ , 𝒳 ε , δ E m 0 - δ ^ ) ≥ cupl ⁢ ( Ψ ε ∘ Φ ε ) ;

moreover, since Ψε∘Φε is homotopic to the immersion j thanks to Proposition 4.12, we have by Lemma 5.3 (b)

cupl ⁢ ( Ψ ε ∘ Φ ε ) = cupl ⁢ ( j ) ,

which leads to the conclusion thanks to Lemma 5.4. See Remark 5.8 for the proof of regularity. ∎

5.1 Concentration in K

We prove now the polynomial decay and the concentration of the found solutions in K. First, we recall some definitions and properties of the fractional De Giorgi class introduced in [25], to which we refer for a complete introduction on the topic; we focus only on the linear case, where the exponent of the space is given by 2.

Set first

Tail ⁢ ( u ; x 0 , R ) := ( 1 - s ) ⁢ R 2 ⁢ s ⁢ ∫ ℝ N ∖ B R ⁢ ( x 0 ) | u ⁢ ( x ) | | x - x 0 | N + 2 ⁢ s ⁢ 𝑑 x ,

which is the tail function of u∈Hs⁢(ℝN), centered in x0∈ℝN with radius R>0, introduced in [29, 30]. From [25, Paragraph 6.1] we have the following definition.

Definition 5.6.

Let A⊂ℝN be open, ζ≥0, H≥1, k0∈ℝ, μ∈(0,2⁢sN], λ≥0 and R0∈(0,+∞]. We say that u belongs to the fractional De Giorgi classD⁢G+s,2⁢(A,ζ,H,k0,μ,λ,R0) if and only if

[ ( u - k ) + ] B r ⁢ ( x 0 ) 2 + ∫ B r ⁢ ( x 0 ) ( u ⁢ ( x ) - k ) + ⁢ ( ∫ B 2 ⁢ R 0 ⁢ ( x ) ( u ⁢ ( y ) - k ) - | x - y | N + 2 ⁢ s ⁢ 𝑑 y ) ⁢ 𝑑 x

≤ H 1 - s ( ( R λ ζ 2 + | k | 2 R N ⁢ μ ) | supp ( ( u - k ) + ) ∩ B R ( x 0 ) | 1 - 2 ⁢ s N + μ + R 2 ⁢ ( 1 - s ) ( R - r ) 2 ∥ ( u - k ) + ∥ L 2 ⁢ ( B R ⁢ ( x 0 ) ) 2

+ R N ( R - r ) N + 2 ⁢ s ∥ ( u - k ) + ∥ L 1 ⁢ ( B R ⁢ ( x 0 ) ) Tail ( ( u - k ) + ; x 0 , r ) )

for any x0∈A, 0<r<R<min⁡{R0,d⁢(x0,∂⁡A)} and k≥k0. Here (u-k)± denote the positive and negative parts of the function u-k.

We see now how this class of functions is related to the PDE setting. By a careful analysis of [25, proof of Proposition 8.5] we obtain the following result.

Theorem 5.7.

Let N≥2 and let u∈Hs⁢(RN) be a weak subsolution of

( - Δ ) s ⁢ u ≤ g ⁢ ( x , u ) , x ∈ ℝ N ,

where g:RN×R→R satisfies, for a.e. x∈RN and every t∈R,

| g ⁢ ( x , t ) | ≤ d 1 + d 2 ⁢ | t | q - 1

for some q∈(2,2s*). Then there exist α=α⁢(N,s,q)>0, C=C⁢(N,s,q,d2)>0 and H=H⁢(N,s,q,d2)≥1 such that, for each x0∈RN and each R0 verifying

0 < R 0 ≤ C ⁢ ( N , s , q , d 2 ) ⁢ min ⁡ { 1 , ∥ u ∥ L 2 s * ⁢ ( ℝ N ) - α ⁢ ( N , s , q ) } ,

it results that

u ∈ DG + s , 2 ⁢ ( B R 0 ⁢ ( x 0 ) , d 1 , H , 0 , 1 - q 2 s * , 2 ⁢ s , R 0 ) .

As shown in [25, Proposition 6.1 and Theorem 8.2], the belonging to a De Giorgi class implies useful Lloc∞ and Cloc0,σ estimates, which will be implemented in the following proof.

Proof of Theorem 1.4.

For ε sufficiently small, let uε be one of the cupl⁢(K)+1 critical points of Jε built in Theorem 1.4, which by Corollary 4.9 is also a solution of (4.1), positive by (f2). In particular, since it satisfies the assumptions of Lemma 4.6, looking at the proof (see (4.34) and (4.33)), we obtain that

∥ u ε - U ( ⋅ - p ε ) ∥ H s ⁢ ( ℝ N ) → 0

with U∈SV⁢(p0), pε∈ℝN and

ε ⁢ p ε → p 0 ∈ Ω ⁢ [ 0 , ν 1 ) .

Step 1. Notice that we have found these solutions by fixing ν0, l0 and l0′. Let them move, throughout three sequences ν0n↘0, l0n↘Em0, and (l0′)n↘Em0, and find the corresponding (sufficiently small) εn>0 such that cupl⁢(K)+1 solutions exist; let uεn be one of those and pεn as before. It is not reductive to assume εn→0 as n→+∞; by a diagonalization-like argument we obtain

(5.2) u ε n ( ⋅ + p ε n ) → U in H s ⁢ ( ℝ N ) for some U least energy solution of (1.6) ,

and

ε n ⁢ p ε n → p 0 ∈ K ,

as n→+∞.

Step 2. From now on we write ε≡εn to avoid cumbersome notation. By Iε′⁢(uε)=0 we obtain

(5.3) ( - Δ ) s ⁢ u ε + V ⁢ ( ε ⁢ x ) ⁢ u ε = f ⁢ ( u ε ) , x ∈ ℝ N ,

thus (recall that uε is positive), by choosing β<V¯ in (2.5),

( - Δ ) s ⁢ u ε ≤ - V ¯ ⁢ u ε + f ⁢ ( u ε ) ≤ ( β - V ¯ ) ⁢ u ε + C β ⁢ u ε p ≤ C β ⁢ u ε p , x ∈ ℝ N .

Therefore by Theorem 5.7 we have, choosing q=p, d1=0 and d2=Cβ,

u ε ∈ DG + s , 2 ⁢ ( B R 0 ⁢ ( x 0 ) , 0 , H , 0 , 1 - p 2 s * , 2 ⁢ s , R 0 ) ,

with H=H⁢(N,s,p,β) and R0 depending on N,s,q,Cβ and a uniform upper bound of the Hs-norms of uε.

We can thus use now [25, Proposition 6.1]: observing that d⁢(x0,∂⁡BR0⁢(x0))=R0, and that μ=1-p2s*, we obtain, for any ω∈(0,1] and R∈(0,R02),

sup B R ⁢ ( x 0 ) ⁡ u ε ≤ C ( N - 2 ⁢ s ) 1 2 ⁢ μ ⁢ 1 ω 1 2 ⁢ μ ⁢ 1 ( 2 ⁢ R ) N / 2 ⁢ ∥ u ε ∥ L 2 ⁢ ( B 2 ⁢ R ⁢ ( x 0 ) ) + ω ⁢ Tail ⁢ ( u ε ; x 0 , R ) ,

that is, rewriting the constant C=C⁢(N,s,p,β),

sup B R ⁢ ( x 0 ) ⁡ u ε ≤ C ⁢ 1 ω 1 2 ⁢ μ ⁢ 1 R N / 2 ⁢ ∥ u ε ∥ L 2 ⁢ ( B 2 ⁢ R ⁢ ( x 0 ) ) + ω ⁢ Tail ⁢ ( u ε ; x 0 , R ) .

Step 3. We have

∥ u ε ∥ L ∞ ⁢ ( ℝ N ) = sup x 0 ∈ ℝ N ⁡ sup B R ⁢ ( x 0 ) ⁡ u ε ≤ sup x 0 ∈ ℝ N ⁡ ( C ⁢ 1 ω 1 2 ⁢ μ ⁢ 1 R N / 2 ⁢ ∥ u ε ∥ L 2 ⁢ ( B 2 ⁢ R ⁢ ( x 0 ) ) + ω ⁢ Tail ⁢ ( u ε ; x 0 , R ) ) .

Observe that, by Hölder inequality,

Tail ⁢ ( u ε ; x 0 , R ) ≤ ( 1 - s ) ⁢ R 2 ⁢ s ⁢ ∥ u ε ∥ L 2 ⁢ ( ℝ N ∖ B R ⁢ ( x 0 ) ) ⁢ ∥ 1 | x - x 0 | 2 ⁢ N + 4 ⁢ s ∥ L 2 ⁢ ( ℝ N ∖ B R ⁢ ( x 0 ) ) ≤ C R N / 2 ⁢ ∥ u ε ∥ L 2 ⁢ ( ℝ N ) .

Thus

∥ u ε ∥ L ∞ ⁢ ( ℝ N ) ≤ C R N / 2 ⁢ sup x 0 ∈ ℝ N ⁡ ( ω - 1 2 ⁢ μ ⁢ ∥ u ε ∥ L 2 ⁢ ( B 2 ⁢ R ⁢ ( x 0 ) ) + ω ⁢ ∥ u ε ∥ L 2 ⁢ ( ℝ N ∖ B R ⁢ ( x 0 ) ) ) ≤ C R N / 2 ⁢ ( ω - 1 2 ⁢ μ + ω ) ⁢ ∥ u ε ∥ L 2 ⁢ ( ℝ N ) ,

which is uniformly bounded by the properties on uε. Hence uε are uniformly bounded in L∞⁢(ℝN).

In addition, by the estimates on V, f and uε, we have

g ε ⁢ ( x ) := - V ⁢ ( ε ⁢ x ) ⁢ u ε ⁢ ( x ) + f ⁢ ( u ε ⁢ ( x ) ) ∈ L ∞ ⁢ ( ℝ N )

with bound uniform in ε; since

( - Δ ) s ⁢ u ε = g ε ⁢ ( x ) , x ∈ ℝ N ,

by [25, Theorem 8.2] there exists σ∈(0,1), not depending on uε, and C=C⁢(N,s), such that, for each R>1 and x0∈ℝN,

[ u ε ] C 0 , σ ⁢ ( B R ⁢ ( x 0 ) ) ≤ C R σ ⁢ ( ∥ u ε ∥ L ∞ ⁢ ( B 4 ⁢ R ⁢ ( x 0 ) ) + Tail ⁢ ( u ε ; x 0 , 4 ⁢ R ) + R 2 ⁢ s ⁢ ∥ g ε ∥ L ∞ ⁢ ( B 8 ⁢ R ⁢ ( x 0 ) ) ) ≤ C ′ .

We highlight that, since the constant is uniform in R>1, we obtain uε∈C0,σ⁢(ℝN).

Step 4. By the local uniform estimate on uε we could gain ∥uε∥L∞⁢(ℝN∖(Ωh0/ε)2⁢R0)→0, but this lack of uniformity on the domain can be improved. Thus we exploit the tightness of u~ε to reach the claim, where

u ~ ε := u ε ( ⋅ + p ε ) .

Indeed, by Step 2, and (5.2) we have

{ ( - Δ ) s ⁢ u ~ ε + V ⁢ ( ε ⁢ x + ε ⁢ p ε ) ⁢ u ~ ε = f ⁢ ( u ~ ε ) , x ∈ ℝ N , ∥ u ~ ε ∥ ∞ ≤ C , u ~ ε → U in H s ⁢ ( ℝ N ) as ε → 0 , U least energy solution of (1.6) .

In particular, it is standard to show that f⁢(u~ε)→f⁢(U) in L2⁢(ℝN), ∥f⁢(u~ε)∥∞≤C and U,f⁢(U)∈L∞⁢(ℝN). By interpolation we thus obtain

χ ε := u ~ ε + f ⁢ ( u ~ ε ) → χ := U + f ⁢ ( U ) in L q ′ ⁢ ( ℝ N )

for every q′∈[2,+∞), and ∥χε∥∞≤C. Proceeding as in [2, Lemma 2.6], we obtain

(5.4) u ~ ε ⁢ ( x ) → 0 as | x | → + ∞ , uniformly in ε .

For the reader’s convenience, we sketch here the proof. Indeed, being u~ε solution of

( - Δ ) s ⁢ u ~ ε + u ~ ε = χ ε - V ⁢ ( ε ⁢ x + ε ⁢ p ε ) ⁢ u ~ ε , x ∈ ℝ N ,

we have the representation formula

u ~ ε = 𝒦 * ( χ ε - V ⁢ ( ε ⁢ x + ε ⁢ p ε ) ⁢ u ~ ε ) ,

where 𝒦 is the Bessel Kernel; we recall that 𝒦 is positive, satisfies 𝒦⁢(x)≤C|x|N+2⁢s for |x|≥1 and 𝒦∈Lq⁢(ℝN) for q∈[1,1+2⁢sN-2⁢s) (see [33, p. 1241 and Theorem 3.3]). Let us fix η>0; since V, u~ε and 𝒦 are positive, we have, for x∈ℝN,

u ~ ε ⁢ ( x ) = ∫ ℝ N 𝒦 ⁢ ( x - y ) ⁢ ( χ ε ⁢ ( y ) - V ⁢ ( ε ⁢ x + ε ⁢ p ε ) ⁢ u ~ ε ⁢ ( y ) ) ⁢ 𝑑 y ≤ ∫ | x - y | ≥ 1 η 𝒦 ⁢ ( x - y ) ⁢ χ ε ⁢ ( y ) ⁢ 𝑑 y + ∫ | x - y | < 1 η 𝒦 ⁢ ( x - y ) ⁢ χ ε ⁢ ( y ) ⁢ 𝑑 y .

As regards the first piece

∫ | x - y | ≥ 1 η 𝒦 ⁢ ( x - y ) ⁢ χ ε ⁢ ( y ) ⁢ 𝑑 y ≤ ∥ χ ε ∥ ∞ ⁢ ∫ | x - y | ≥ 1 / η C | x - y | N + 2 ⁢ s ⁢ 𝑑 y ≤ C ⁢ η 2 ⁢ s ,

while for the second piece, fixed a whatever q∈(1,1+2⁢sN-2⁢s) and its conjugate exponent q′>N2⁢s, we have by Hölder inequality

∫ | x - y | < 1 η 𝒦 ⁢ ( x - y ) ⁢ χ ε ⁢ ( y ) ⁢ 𝑑 y ≤ ∥ 𝒦 ∥ q ⁢ ∥ χ ε ∥ L q ′ ⁢ ( B 1 / η ⁢ ( x ) ) ≤ ∥ 𝒦 ∥ q ⁢ ( ∥ χ ε - χ ∥ q ′ + ∥ χ ∥ L q ′ ⁢ ( B 1 / η ⁢ ( x ) ) ) ,

where the first norm can be made small for ε<ε0=ε0⁢(η), while the second small (uniformly in ε) by |x|≫0. On the other hand, for ε≥ε0 (and thus for a finite number of elements, since we recall we are working with ε≡εn small) the quantity ∥χε∥Lq′⁢(B1/η⁢(x)) can be made small for |x|≫0, uniformly in ε. Joining the pieces, we have (5.4).

Step 5. Let now yε∈ℝN be a maximum point for uε, which exists by the boundedness of uε and its continuity (see (5.1)). Therefore zε:=yε-pε is a maximum point for u~ε. In particular,

u ~ ε ⁢ ( z ε ) = max ℝ N ⁡ u ~ ε = ∥ u ~ ε ∥ ∞ ↛ 0 as ε → 0

since in this case we would have u~ε→0 almost everywhere, which is in contradiction with the fact that u~ε→U≢0 almost everywhere (up to a subsequence). As a consequence, thanks to (5.4), we have that zε is bounded (up to a subsequence). That is, again up to a subsequence,

z ε → p ¯

for some p¯∈ℝN. In particular,

ε ⁢ y ε = ε ⁢ z ε + ε ⁢ p ε → p 0 ∈ K

and, by the fact that

U ( ⋅ + z ε ) → U ( ⋅ + p ¯ ) = : U ¯ in H s ⁢ ( ℝ N )

we have uε(⋅+yε)→U¯ in Hs⁢(ℝN), U¯ least energy solution of (1.6). We set

u ¯ ε := u ε ( ⋅ + y ε ) ,

and note that

u ¯ ε → U ¯ in H s ⁢ ( ℝ N ) , U ¯ least energy solution of (1.6) ;

in addition, u¯ε is positive by (f2), and in the same way we obtained (5.4) we obtain also

(5.5) u ¯ ε ⁢ ( x ) → 0 as | x | → + ∞ , uniformly in ε .

Moreover, by exploiting the uniform estimates in L∞⁢(ℝN) and Cloc0,σ⁢(ℝN) we obtain by the Ascoli–Arzelà theorem also that u¯ε→U¯>0 in Lloc∞⁢(ℝN), with U¯ continuous; this easily implies, for every r>0, that

(5.6) min B r ⁡ u ¯ ε ≥ 1 2 ⁢ min B r ⁡ U ¯ > 0

for ε small, depending on U¯ and r.

Step 6. By (5.5) we have, for R′ large (uniform in ε), that

u ¯ ε ⁢ ( x ) ≤ η ′ for | x | > R ′

for every ε>0, where η′>0 is preliminary fixed. As a consequence, by (f1.2), we gain

- 1 2 ⁢ V ¯ ⁢ u ¯ ε ⁢ ( x ) ≤ f ⁢ ( u ¯ ε ⁢ ( x ) ) ≤ 1 2 ⁢ V ¯ ⁢ u ¯ ε ⁢ ( x ) for | x | > R ′ ,

where V¯:=∥V∥∞. We obtain by (5.3)

( - Δ ) s ⁢ u ¯ ε + 1 2 ⁢ V ¯ ⁢ u ¯ ε ≤ f ⁢ ( u ¯ ε ) - 1 2 ⁢ V ¯ ⁢ u ¯ ε ≤ 0 , x ∈ ℝ N ∖ B R ′ ,

( - Δ ) s ⁢ u ¯ ε + 3 2 ⁢ V ¯ ⁢ u ¯ ε ≥ f ⁢ ( u ¯ ε ) + 1 2 ⁢ V ¯ ⁢ u ¯ ε ≥ 0 , x ∈ ℝ N ∖ B R ′ .

Notice that we always intend differential inequalities in the weak sense, that is, tested with functions in Hs⁢(ℝN) with supports contained (e.g.) in ℝN∖BR′.

In addition, by Lemma A.3 we have that there exist two positive functions W¯′, W¯′ and three positive constants R′′, C′ and C′′ depending only on V, such that

{ ( - Δ ) s ⁢ W ¯ ′ + 3 2 ⁢ V ¯ ⁢ W ¯ ′ = 0 for ⁢ x ∈ ℝ N ∖ B R ′′ , C ′ | x | N + 2 ⁢ s < W ¯ ′ ⁢ ( x ) for ⁢ | x | > 2 ⁢ R ′′ ,

and

{ ( - Δ ) s ⁢ W ¯ ′ + 1 2 ⁢ V ¯ ⁢ W ¯ ′ = 0 for ⁢ x ∈ ℝ N ∖ B R ′′ , W ¯ ′ ⁢ ( x ) < C ′′ | x | N + 2 ⁢ s for | x | > 2 ⁢ R ′′ .

Set R:=max⁡{R′,2⁢R′′}. Let C¯1 and C¯1 be some uniform lower and upper bounds for u¯ε on BR, C¯2:=minBR⁡W¯′ and C¯2:=maxBR⁡W¯′, all strictly positive. Define

W ¯ := C ¯ 1 ⁢ C ¯ 2 - 1 ⁢ W ¯ ′ , W ¯ := C ¯ 1 ⁢ C ¯ 2 - 1 ⁢ W ¯ ′

so that

W ¯ ≤ u ¯ ε ≤ W ¯ for | x | ≤ R .

Through a Comparison Principle (see Lemma A.1), and redefining C′ and C′′, we obtain

C ′ | x | N + 2 ⁢ s < W ¯ ⁢ ( x ) ≤ u ¯ ε ⁢ ( x ) ≤ W ¯ ⁢ ( x ) < C ′′ | x | N + 2 ⁢ s for | x | > R .

By the uniform boundedness of u¯ε and (5.6) we also obtain

C ′ 1 + | x | N + 2 ⁢ s < u ¯ ε ⁢ ( x ) < C ′′ 1 + | x | N + 2 ⁢ s for x ∈ ℝ N .

Recalling the definition of u¯ε, we have finally obtained a sequence of solutions such that

{ u ε n ⁢ ( y ε n ) = max ℝ N ⁡ u ε n , d ⁢ ( ε n ⁢ y ε n , K ) → 0 , C ′ 1 + | x - y ε n | N + 2 ⁢ s ≤ u ε n ⁢ ( x ) ≤ C ′′ 1 + | x - y ε n | N + 2 ⁢ s for x ∈ ℝ N , ∥ u ε n ( ⋅ + y ε n ) - U ¯ ∥ H s ⁢ ( ℝ N ) → 0 for some U ¯ least energy solution of (1.6) ,

where the limits are given by n→+∞. Furthermore, by the uniform estimates in L∞⁢(ℝN) and the local uniform estimates in Cloc0,σ⁢(ℝN) of uεn, together with the locally-compact version of the Ascoli–Arzelà theorem, we have that the last convergence is indeed uniform on compacts. Thus, recalled that vεn=uεn⁢(⋅εn) are solutions of the original problem (1.2), defined xεn:=εn⁢yεn we obtain, as n→+∞,

{ v ε n ⁢ ( x ε n ) = max ℝ N ⁡ v ε n , d ⁢ ( x ε n , K ) → 0 , C ′ 1 + | x - x ε n ε n | N + 2 ⁢ s ≤ v ε n ⁢ ( x ) ≤ C ′′ 1 + | x - x ε n ε n | N + 2 ⁢ s for x ∈ ℝ N , ∥ v ε n ( ε n ⋅ + x ε n ) - U ¯ ∥ X → 0 , X = H s ( ℝ N ) and X = L loc ∞ ( ℝ N ) ,

for some U¯ least energy solution of (1.6). This concludes the proof. ∎

Remark 5.8.

We observe that Steps 2 and 3 apply to a whatever family of equations (uε)ε, that is, why the regularity statement in Theorem 1.1 holds true.

Communicated by Antonio Ambrosetti

Funding statement: The authors are supported by MIUR-PRIN project “Qualitative and quantitative aspects of nonlinear PDEs” (2017JPCAPN_005), and partially supported by GNAMPA-INdAM.

A Polynomial Decay of S^

We recall some lemmas useful to prove polynomial decay estimates. We start with a comparison principle: the result is a straightforward adaptation of [47, Lemma 6], and we prove it here for the sake of completeness. See also [38].

Lemma A.1 (Maximum Principle).

Let Σ⊂RN, possibly unbounded, and let u∈Hs⁢(RN) be a weak subsolution of

( - Δ ) s ⁢ u + a ⁢ u ≤ 0 , x ∈ ℝ N ∖ Σ

with a>0, in the sense that

∫ ℝ N ( - Δ ) s 2 ⁢ u ⁢ ( - Δ ) s 2 ⁢ v ⁢ 𝑑 x + a ⁢ ∫ ℝ N u ⁢ v ⁢ 𝑑 x ≤ 0

for every positive v∈Hs⁢(RN) with supp(v)⊂RN∖Σ. Assume moreover that

u ≤ 0 for a.e. x ∈ Σ .

Then

(A.1) u ≤ 0 for a.e. x ∈ ℝ N .

Proof.

By the assumption we have u+=0 on Σ, thus u+∈Hs⁢(ℝN) is a suitable test function and we obtain, using u=u+-u- and u+⁢u-≡0,

0 ≥ ∫ ℝ N | ( - Δ ) s 2 ⁢ u + | 2 ⁢ 𝑑 x + a ⁢ ∫ ℝ N | u + | 2 ⁢ 𝑑 x - ∫ ℝ N ( - Δ ) s 2 ⁢ u - ⁢ ( - Δ ) s 2 ⁢ u + ⁢ 𝑑 x

= ∥ ( - Δ ) s 2 ⁢ u + ∥ 2 2 + a ⁢ ∥ u + ∥ 2 2 + C ⁢ ∫ ℝ 2 ⁢ N u - ⁢ ( x ) ⁢ u + ⁢ ( y ) + u - ⁢ ( y ) ⁢ u + ⁢ ( x ) | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y

≥ ∥ ( - Δ ) s 2 ⁢ u + ∥ 2 2 + a ⁢ ∥ u + ∥ 2 2 ,

which implies u+=0. ∎

Remark A.2.

We point out that if u is assumed continuous, then (A.1) is actually pointwise. Moreover, the constant a>0 may be substituted by a more general a⁢(x)>0 which gives sense to the integrals.

The following result can be obtained similarly to [33, Lemmas 4.2 and 4.3].

Lemma A.3 (Comparison Function).

Let b>0. There exists a strictly positive continuous function Wb∈Hs⁢(RN) such that, for some positive constants Cb′,Cb′′, it verifies

( - Δ ) s ⁢ W b + b 2 ⁢ W b = 0 , x ∈ ℝ N ∖ B r b

pointwise, with rb:=(2b)12⁢s, and

C b ′ | x | N + 2 ⁢ s < W b ⁢ ( x ) < C b ′′ | x | N + 2 ⁢ s for | x | > 2 ⁢ r b .

The constants rb,Cb′,Cb′′ remain bounded by letting b vary in a compact set far from zero.

Proof.

Let B1/2≺φ≺B1, and define W~:=𝒦*φ, where 𝒦 is the Bessel potential. Arguing as in [33] (see also [16, Theorem 1.3]), we obtain

( - Δ ) s ⁢ W ~ + W ~ = φ , x ∈ ℝ N ,

and

C ′ | x | N + 2 ⁢ s < W ~ ⁢ ( x ) ≤ C ′′ | x | N + 2 ⁢ s .

By scaling W:=W~(rb⋅), we reach the claim. ∎

Proposition A.4 (Polynomial Decay).

Assume (f1)–(f3). Let a>0 and let U be a weak solution of

( - Δ ) s ⁢ U + a ⁢ U = f ⁢ ( U ) , x ∈ ℝ N .

Then there exist positive constants Ca′,Ca′′ such that

C a ′ 1 + | x | N + 2 ⁢ s ≤ U ⁢ ( x ) ≤ C a ′′ 1 + | x | N + 2 ⁢ s for x ∈ ℝ N .

These constants can be chosen uniform for U∈S^.

Proof.

The proof is almost the same of the one carried out in Theorem 1.4. Indeed, as in Step 2 and Step 3 of that proof, we obtain the uniform boundedness in L∞⁢(ℝN). We point out that the values Cδ, H, C and R0 depend on a∈[m0,m0+ν0], since they depend on δ and we must have δ<a; on the other hand, it is sufficient to take δ<m0 to gain uniformity. The same can be said on the uniform boundedness in C0,σ⁢(ℝN) and for the constants R′′,C′,C′′ related to the comparison functions W¯′,W¯′, thanks to Lemma A.3. As we will show, this allows us to gain that

(A.2) lim | x | → + ∞ ⁡ U ⁢ ( x ) = 0 uniformly for U ∈ S ^ ,

which leads, as in Step 6 of the proof, to

| f ⁢ ( U ⁢ ( x ) ) | ≤ 1 2 ⁢ a ⁢ U ⁢ ( x ) for | x | > R ′ ,

where R′ does not depend on a∈[m0,m0+ν0]. In addition, compactness of S^ and a simple contradiction argument lead to minBR⁡U≥C>0 uniformly for U∈S^. If we prove (A.2), we conclude as in Step 6.

Let us prove (A.2). By contradiction, there exist (xk)k⊂ℝN, |xk|→+∞, (Uk)k⊂S^ and θ>0 such that Uk⁢(xk)>θ>0. Define

V k := U k ( ⋅ + x k ) .

Since both are bounded sequences in Hs⁢(ℝN), we have Uk⇀U and Vk⇀V in Hs⁢(ℝN); moreover, by the uniform L∞⁢(ℝN) and Cloc0,σ⁢(ℝN) estimates and the Ascoli–Arzelà theorem, we have also that the convergences are pointwise. In particular, by

U k ⁢ ( 0 ) ≥ U k ⁢ ( x k ) > θ , V k ⁢ ( 0 ) = U k ⁢ ( x k ) > θ

we obtain

U ⁢ ( 0 ) ≥ θ > 0 , V ⁢ ( 0 ) ≥ θ > 0 .

As a consequence, U and V are not trivial. Let now (ak)k⊂ℝ be such that Uk∈Sak; up to a subsequence we have ak→a∈[m0,m0+ν0]. Observed that also Vk are solutions of Lak′⁢(Vk)=0, we obtain, as in Step 3 of the proof of Lemma 3.4 (see also Step 7 of the proof of Lemma 4.6), that U and V are (nontrivial) solutions of La′⁢(U)=0. Hence

E m 0 ≤ E a ≤ L a ⁢ ( U ) , E m 0 ≤ E a ≤ L a ⁢ ( V ) .

By the Pohozaev identity we have the following chain of inequalities, once fixed R>0 and k≫0 such that |xk|≥2⁢R,

l 0 ≥ lim inf k → + ∞ ⁡ L a k ⁢ ( U k )

= s N ⁢ lim inf k → + ∞ ⁡ ∫ ℝ N | ( - Δ ) s 2 ⁢ U k | 2 ⁢ 𝑑 x

≥ s N ⁢ lim inf k → + ∞ ⁡ ( ∫ B R | ( - Δ ) s 2 ⁢ U k | 2 ⁢ 𝑑 x + ∫ B R | ( - Δ ) s 2 ⁢ V k | 2 ⁢ 𝑑 y )

≥ s N ⁢ ( ∫ B R | ( - Δ ) s 2 ⁢ U | 2 ⁢ 𝑑 x + ∫ B R | ( - Δ ) s 2 ⁢ V | 2 ⁢ 𝑑 y ) ,

where in the last passage we have used that Uk⇀U in Hs⁢(ℝN), thus (-Δ)s2⁢Uk⇀(-Δ)s2⁢U in L2⁢(ℝN)↪L2⁢(BR), and the weak lower semicontinuity of the norm. Thus, by choosing R sufficiently large, we have

l 0 ≥ s N ⁢ ( ∫ ℝ N | ( - Δ ) s 2 ⁢ U | 2 ⁢ 𝑑 x + ∫ ℝ N | ( - Δ ) s 2 ⁢ V | 2 ⁢ 𝑑 y ) - η = L a ⁢ ( U ) + L a ⁢ ( V ) - η ≥ 2 ⁢ E m 0 - η ,

which leads to a contradiction if we choose η∈(0,2⁢Em0-l0), possible thanks to (3.4). ∎

Acknowledgements

The authors would like also to thank Prof. K. Tanaka for the fruitful comments.

References

[1] C. O. Alves and V. Ambrosio, A multiplicity result for a nonlinear fractional Schrödinger equation in ℝN without the Ambrosetti–Rabinowitz condition, J. Math. Anal. Appl. 466 (2018), no. 1, 498–522. 10.1016/j.jmaa.2018.06.005Search in Google Scholar

[2] C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in ℝN via penalization method, Calc. Var. Partial Differential Equations 55 (2016), no. 3, Article ID 47. 10.1007/s00526-016-0983-xSearch in Google Scholar

[3] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 140 (1997), no. 3, 285–300. 10.1007/s002050050067Search in Google Scholar

[4] A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal. 159 (2001), no. 3, 253–271. 10.1007/s002050100152Search in Google Scholar

[5] V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in ℝN, Rev. Mat. Iberoam. 35 (2019), no. 5, 1367–1414. 10.4171/rmi/1086Search in Google Scholar

[6] A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), no. 3, 253–294. 10.1002/cpa.3160410302Search in Google Scholar

[7] T. Bartsch, Topological Methods for Variational Problems with Symmetries, Lecture Notes in Math. 1560, Springer, Berlin, 1993. 10.1007/BFb0073859Search in Google Scholar

[8] T. Bartsch, Note on category and cup-length, private communication 2012. Search in Google Scholar

[9] T. Bartsch and T. Weth, The effect of the domain’s configuration space on the number of nodal solutions of singularly perturbed elliptic equations, Topol. Methods Nonlinear Anal. 26 (2005), no. 1, 109–133. 10.12775/TMNA.2005.027Search in Google Scholar

[10] V. Benci and G. Cerami, Existence of positive solutions of the equation -Δ⁢u+a⁢(x)⁢u=u(N+2)/(N-2) in 𝐑N, J. Funct. Anal. 88 (1990), no. 1, 90–117. 10.1016/0022-1236(90)90120-ASearch in Google Scholar

[11] V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Ration. Mech. Anal. 114 (1991), no. 1, 79–93. 10.1007/BF00375686Search in Google Scholar

[12] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313–345. 10.1007/BF00250555Search in Google Scholar

[13] U. Biccari, M. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud. 17 (2017), no. 2, 387–409. 10.1515/ans-2017-0014Search in Google Scholar

[14] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal. 185 (2007), no. 2, 185–200. 10.1007/s00205-006-0019-3Search in Google Scholar

[15] J. Byeon and L. Jeanjean, Erratum: “Standing waves for nonlinear Schrödinger equations with a general nonlinearity” [Arch. Ration. Mech. Anal. 185 (2007), no. 2, 185–200; mr2317788], Arch. Ration. Mech. Anal. 190 (2008), no. 3, 549–551. 10.1007/s00205-008-0178-5Search in Google Scholar

[16] J. Byeon, O. Kwon and J. Seok, Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity 30 (2017), no. 4, 1659–1681. 10.1088/1361-6544/aa60b4Search in Google Scholar

[17] J. Byeon and K. Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1859–1899. 10.4171/JEMS/407Search in Google Scholar

[18] J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Mem. Amer. Math. Soc. 229 (2014), no. 1076, 1–89. Search in Google Scholar

[19] K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993. 10.1007/978-1-4612-0385-8Search in Google Scholar

[20] G. Chen, Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations, Nonlinearity 28 (2015), no. 4, 927–949. 10.1088/0951-7715/28/4/927Search in Google Scholar

[21] S. Cingolani, L. Jeanjean and K. Tanaka, Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well, Calc. Var. Partial Differential Equations 53 (2015), no. 1–2, 413–439. 10.1007/s00526-014-0754-5Search in Google Scholar

[22] S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal. 10 (1997), no. 1, 1–13. 10.12775/TMNA.1997.019Search in Google Scholar

[23] S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 5, 973–1009. 10.1017/S0308210509000584Search in Google Scholar

[24] S. Cingolani and K. Tanaka, Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well, Rev. Mat. Iberoam. 35 (2019), no. 6, 1885–1924. 10.4171/rmi/1105Search in Google Scholar

[25] M. Cozzi, Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes, J. Funct. Anal. 272 (2017), no. 11, 4762–4837. 10.1016/j.jfa.2017.02.016Search in Google Scholar

[26] J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations 256 (2014), no. 2, 858–892. 10.1016/j.jde.2013.10.006Search in Google Scholar

[27] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), no. 2, 121–137. 10.1007/BF01189950Search in Google Scholar

[28] M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal. 149 (1997), no. 1, 245–265. 10.1006/jfan.1996.3085Search in Google Scholar

[29] A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), no. 6, 1807–1836. 10.1016/j.jfa.2014.05.023Search in Google Scholar

[30] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 5, 1279–1299. 10.1016/j.anihpc.2015.04.003Search in Google Scholar

[31] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar

[32] M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity 28 (2015), no. 6, 1937–1961. 10.1088/0951-7715/28/6/1937Search in Google Scholar

[33] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 6, 1237–1262. 10.1017/S0308210511000746Search in Google Scholar

[34] G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick–Schnirelmann category and Morse theory for a fractional Schrödinger equation in ℝN, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 2, Article ID 12. 10.1007/s00030-016-0355-4Search in Google Scholar

[35] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), no. 3, 397–408. 10.1016/0022-1236(86)90096-0Search in Google Scholar

[36] G. Fournier, D. Lupo, M. Ramos and M. Willem, Limit relative category and critical point theory, Dynamics Reported. Expositions in Dynamical Systems, Springer, Berlin (1994), 1–24. 10.1007/978-3-642-78234-3_1Search in Google Scholar

[37] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671–1726. 10.1002/cpa.21591Search in Google Scholar

[38] S. Jarohs, Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings, Adv. Nonlinear Stud. 18 (2018), no. 4, 691–704. 10.1515/ans-2017-6039Search in Google Scholar

[39] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), no. 4–6, 298–305. 10.1016/S0375-9601(00)00201-2Search in Google Scholar

[40] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3) 66 (2002), no. 5, Article ID 056108. 10.1142/9789813223806_0003Search in Google Scholar

[41] W. S. Massey, A Basic Course in Algebraic Topology, Grad. Texts in Math. 127, Springer, New York, 1991. 10.1007/978-1-4939-9063-4Search in Google Scholar

[42] Y.-G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V)a, Comm. Partial Differential Equations 13 (1988), no. 12, 1499–1519. 10.1080/03605308808820585Search in Google Scholar

[43] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), no. 2, 270–291. 10.1007/BF00946631Search in Google Scholar

[44] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ℝN, J. Math. Phys. 54 (2013), no. 3, Article ID 031501. 10.1063/1.4793990Search in Google Scholar

[45] S. Secchi and M. Squassina, Soliton dynamics for fractional Schrödinger equations, Appl. Anal. 93 (2014), no. 8, 1702–1729. 10.1080/00036811.2013.844793Search in Google Scholar

[46] J. Seok, Spike-layer solutions to nonlinear fractional Schrödinger equations with almost optimal nonlinearities, Electron. J. Differential Equations 2015 (2015), Article No. 196. Search in Google Scholar

[47] R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat. 58 (2014), no. 1, 133–154. 10.5565/PUBLMAT_58114_06Search in Google Scholar

[48] A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals, Nonlinear Anal. 15 (1990), no. 8, 725–739. 10.1016/0362-546X(90)90089-YSearch in Google Scholar

[49] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys. 153 (1993), no. 2, 229–244. 10.1007/BF02096642Search in Google Scholar

Received: 2020-10-05

Revised: 2020-11-02

Accepted: 2020-11-02

Published Online: 2020-11-12

Published in Print: 2021-02-01

On the Fractional NLS Equation and the Effects of the Potential Well’s Topology

Abstract

1 Introduction

Theorem 1.1.

Remark 1.2.

Remark 1.3.

Theorem 1.4.

2 Preliminaries

Further Notation.

3 Limiting Equation

3.1 A Single Equation

Theorem 3.1 (Existence of a Pohozaev Minimum).

Proposition 3.2 (Regularity and Polynomial Decay).

Lemma 3.3.

Proof.

3.2 A Family of Equations

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

Lemma 3.6.

Proof.

3.3 Fractional Center of Mass

Lemma 3.7.

Proof.

4 Singularly Perturbed Equation

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Remark 4.3.

4.1 A Penalized Functional

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

4.2 Critical Points and Palais–Smale Condition

Lemma 4.6.

Theorem 4.7.

Remark 4.8.

Proof of Lemma 4.6.

Corollary 4.9.

Proof.

Proposition 4.10.

Proof.

4.3 Deformation Lemma on a Neighborhood of Expected Solutions

Lemma 4.11.

Proof.

4.4 Maps Homotopic to the Embedding

4.4.1 Definition of Φε

4.4.2 Definition of Ψε

4.4.3 An Homotopy to the Identity

Proposition 4.12.

Proof.

5 Existence of Multiple Solutions

Remark 5.1.

Definition 5.2.

Lemma 5.3.

Lemma 5.4.

Lemma 5.5.

Proof of Theorem 1.1.

5.1 Concentration in K

Definition 5.6.

Theorem 5.7.

Proof of Theorem 1.4.

Remark 5.8.

A Polynomial Decay of S^

Lemma A.1 (Maximum Principle).

Proof.

Remark A.2.

Lemma A.3 (Comparison Function).

Proof.

Proposition A.4 (Polynomial Decay).

Proof.

Acknowledgements

References

Journal and Issue

Articles in the same Issue