The concentration-compactness principles for Ws,p(·,·)(ℝN) and application

Ky Ho; Yun-Ho Kim

doi:10.1515/anona-2020-0160

Open Access Published by De Gruyter December 8, 2020

The concentration-compactness principles for W^s,p(·,·)(ℝ^N) and application

Ky Ho and Yun-Ho Kim

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0160

Abstract

We obtain a critical imbedding and then, concentration-compactness principles for fractional Sobolev spaces with variable exponents. As an application of these results, we obtain the existence of many solutions for a class of critical nonlocal problems with variable exponents, which is even new for constant exponent case.

Keywords: Fractional p-Laplacian; p(·)-Laplacian; fractional Sobolev spaces with variable exponents; critical imbedding; concentration-compactness principles; variational methods

MSC 2010: 35B33; 35D30; 35J20; 35R11; 46E35; 49J35

1 Introduction

Nonlocal equations have been modeled for various problems in real fields, for instance, phase transitions, thin obstacle problem, soft thin films, crystal dislocation, stratified materials, anomalous diffusion, semipermeable membranes and flame propagation, material science, ultra-relativistic limits of quantum mechanics, multiple scattering, minimal surfaces, water waves, etc. After the seminal papers by Caffarelli et al. [1, 2, 3], problems involving fractional P-Laplacian have been intensively studied. On the other hand, various other real fields such as electrorheological fluids and image processing, etc. require partial differential equations with variable exponents (see e.g., [4, 5]). Natural solution spaces for those problems are Sobolev spaces with fractional order or variable exponents, which were comprehensively investigated in [6] and [7].

Recently, many authors have been studied the fractional Sobolev spaces with variable exponents and the corresponding nonlocal equations with variable exponents (see e.g., [11, 9, 8, 10]). To the authors’ best knowledge, though most properties of the classical fractional Sobolev spaces have been extended to the fractional Sobolev spaces with variable exponents, there have no results for the critical Sobolev type imbedding for these spaces. Consequently, there have no results on nonlocal equations with variable critical growth because the critical Sobolev type imbedding is essential in the study of such critical equations. The critical problem was initially studied in the seminal paper by Brezis-Nirenberg [12], which treated for Laplace equations. Since then there have been extensions of [12] in many directions. Elliptic equations involving critical growth are delicate due to the lack of compactness arising in connection with the variational approach. For such problems, the concentration-compactness principles (the CCPs, for short) introduced by P.L. Lions [13, 14] and its variant at infinity [16, 15, 17] have played a decisive role in showing a minimizing sequence or a Palais-Smale sequence is precompact. By using these CCPs or extending them to the Sobolev spaces with fractional order or variable exponents, many authors have been successful to deal with critical problems involving p-Laplacian or p(.)-Laplacian or fractional p-Laplacian, see e.g., [18, 25, 21, 22, 20, 19, 23, 24, 26, 27] and references therein.

As we mentioned above, there have no results for the critical Sobolev type imbedding for the fractional Sobolev spaces with variable exponents. Although the usual critical Sobolev immersion theorem holds in the fractional order or variable exponents setting, we do not know this assertion even in fractional Sobolev spaces with variable exponents defined in bounded domain; see [11, 9, 8, 10]. Because of this, our first aim of the present paper is to obtain a critical imbedding from fractional Sobolev spaces with variable exponents into Lebesgue spaces with variable exponents. We provide sufficient conditions on the variable exponents such as the log-Hölder type continuity condition to obtain such critical imbedding (Theorem 3.3). Thanks to this critical Sobolev imbedding, inspired by [14, 21, 22, 19, 27], we then establish two Lions type concentration-compactness principles for fractional Sobolev spaces with variable exponents, which are our second aim (Theorems 4.1 and 4.2). As an application of these results, we will obtain the existence of many solutions for the following nonlocal problem with variable exponents

(1.1) L u ( x ) + | u | p ( x , x ) − 2 u = f ( x , u ) + λ | u | q ( x ) − 2 u in R N ,

where the operator 𝓛 is defined as

(1.2) L u ( x ) = 2 lim ε ↘ 0 ∫ y ∈ R N : | y − x | ≥ ε | u ( x ) − u ( y ) | p ( x , y ) − 2 ( u ( x ) − u ( y ) ) | x − y | N + s p ( x , y ) d y , x ∈ R N ,

where s ∈ ( 0 , 1 ) , p ∈ C ( R N × R N ) is symmetric i.e., p(x,y)=p(y,x) for all (x,y)∈ℝN×ℝN such that 1 < p − := inf ( x , y ) ∈ R N × R N p ( x , y ) ≤ p + := sup ( x , y ) ∈ R N × R N p ( y , x ) < N s ; q ∈ C R N satisfies p ( x , x ) < q ( x ) ≤ p s ⋆ ( x ) := N p ( x , x ) N − s p ( x , x ) for all x∈ℝN ; 𝜆 is a positive real parameter; and f : R N × R → R is a Carathéodory function of local p+ -superlinear and to be specified later.

The main feature of our final consequence in the present paper is to establish the multiplicity result for problem (1.1) under the critical growth condition x ∈ R N : q ( x ) = p s ⋆ ( x ) ≠ ϕ , originally introduced in [21] for the p(·)-Laplacian case, and some conditions on f different from the related works [18, 28, 29] (Theorem 5.1). As far as we are aware, there are no existence results about the critical problems in this situation even in the case of constant exponents.

The rest of our paper is organized as follows. In Section 2, we briefly review some properties of the Sobolev spaces with fractional order or variable exponents. In Section 3, we establish a critical Sobolev type imbedding for the fractional Sobolev spaces with variable exponents, which is a key to our arguments. In Section 4 we establish Lions type concentration-compactness principles for fractional Sobolev spaces with variable exponents. In Section 5, we show the existence of many solutions for a superlinear nonlocal problem with variable exponents using genus theory. In Appendix, we give an auxiliary result, which is used to prove our CCPs.

2 Variable exponent Lebesgue spaces and fractional Sobolev spaces

In this section, we briefly review the Lebesgue spaces with variable exponents and the classical fractional Sobolev spaces.

Let Ω be a Lipschitz domain in ℝ^N Denote

C + ( Ω ¯ ) = h ∈ C ( Ω ¯ ) : 1 < inf x ∈ Ω ¯ h ( x ) ≤ sup x ∈ Ω ¯ h ( x ) < ∞ ,

and for h ∈ C + ( Ω ¯ ) , denote

h + = sup x ∈ Ω ¯ h ( x ) and h − = inf x ∈ Ω ¯ h ( x ) .

For p ∈ C + ( Ω ¯ ) and a σ-finite, complete measure 𝜇 in Ω, define the variable exponent Lebesgue space L μ p ( ⋅ ) ( Ω ) as

L μ p ( ⋅ ) ( Ω ) := u : Ω → R is μ − measurable, ∫ Ω | u ( x ) | p ( x ) d μ < ∞

endowed with the Luxemburg norm

∥ u ∥ L μ p ( ⋅ ) ( Ω ) := inf λ > 0 : ∫ Ω u ( x ) λ p ( x ) d μ ≤ 1 .

When 𝜇 is the Lebesgue measure, we write d x , L p ( ⋅ ) ( Ω ) and | | u | | L p ( ⋅ ) ( Ω ) instead of 𝜇, d μ , L μ p ( ⋅ ) ( Ω ) and ∥ u ∥ L μ p ( − ) ( Ω ) , respectively. Set L + p ( ⋅ ) ( Ω ) := u ∈ L p ( ⋅ ) ( Ω ) : u > 0 a.e. in Ω and for a Lebesgue measurable and positive a.e. function w : Ω → R , set L p ( ⋅ ) ( w , Ω ) := L μ p ( ⋅ ) ( Ω ) with d μ = w ( x ) d x . Some basic properties of L μ p ( ⋅ ) ( Ω ) are listed in the next three propositions.

Proposition 2.1

([7, Corollary 3.3.4]). Let α , β ∈ C + ( Ω ¯ ) such that α ( x ) ≤ β ( x ) for all x ∈ Ω ¯ . Then, we have

| | u | | L μ α ( ⋅ ) ( Ω ) ≤ 2 [ 1 + μ ( Ω ) ] | | u | | L μ β ( ⋅ ) ( Ω ) , ∀ u ∈ L μ α ( ⋅ ) ( Ω ) ∩ L μ β ( ⋅ ) ( Ω ) .

Proposition 2.2

([30]). Define the modular ρ : L μ p ( ⋅ ) ( Ω ) → R as

ρ ( u ) := ∫ Ω | u | p ( x ) d μ , ∀ u ∈ L p ( ⋅ ) ( Ω )

Then, we have the following relations between the norm and modular.

For u ∈ L μ p ( ⋅ ) ( Ω ) ∖ { 0 } , λ = ∥ u ∥ L μ p ( ⋅ ) ( Ω ) if and only if ρ ( u λ ) = 1.
ρ ( u ) > 1 ( = 1 ; < 1 ) if and only if ∥ u ∥ L μ p ( ⋅ ) ( Ω ) > 1 ( = 1 ; < 1 ) , respectively.
If ∥ u ∥ L μ p ( ⋅ ) ( Ω ) > 1 , then ∥ u ∥ L μ p ( ⋅ ) ( Ω ) p − ≤ ρ ( u ) ≤ ∥ u ∥ L μ p ( ⋅ ) ( Ω ) ∗ p + .
If ∥ u ∥ L μ p ( ⋅ ) ( Ω ) < 1 , then ∥ u ∥ L μ p ( ⋅ ) ( Ω ) p − ≤ ρ ( u ) ≤ ∥ u ∥ L μ p ( ⋅ ) ( Ω ) ∗ p + .

Proposition 2.3

([30, 31]). The space L p ( ⋅ ) ( Ω ) is a separable, uniformly convex Banach space, and its dual space is L p ′ ( ⋅ ) ( Ω ) where 1 / p ( x ) + 1 / p ′ ( x ) = 1 . For any u ∈ L p ( ⋅ ) ( Ω ) and v ∈ L p ′ ( ⋅ ) ( Ω ) , we have

∫ Ω u v d x ≤ 2 ∥ u ∥ L p ( ⋅ ) ( Ω ) ∥ v ∥ L p ′ ( ⋅ ) ( Ω ) .

Let s ∈ (0,1) and p∈(1,∞) be constants. Define the fractional Sobolev space Ws,p(Ω) as

W s , p ( Ω ) := u ∈ L p ( Ω ) : ∫ Ω ∫ Ω | u ( x ) − u ( y ) | p | x − y | N + s p d x d y < ∞

endowed with norm

∥ u ∥ s , p , Ω := ∫ Ω | u ( x ) | p d x + ∫ Ω ∫ Ω | u ( x ) − u ( y ) | p | x − y | N + s p d x d y 1 / p .

We recall the following crucial imbeddings.

Proposition 2.4

([6]). Let s ∈ (0,1) and p∈(1,∞) be such that sp<N. It holds that

W s , p ( Ω ) ↪↪ L q ( Ω ) if Ω is bounded and 1≤q<NpN−sp=:ps∗ ;
W s , p ( Ω ) ↪ L q ( Ω ) if p ≤ q ≤ p s ∗ .

3 The Sobolev spaces W s , p ( ⋅ , ⋅ ) ( Ω )

In this section, we recall the fractional Sobolev spaces with variable exponents that was first introduced in [11], and was then refined in [10]. Furthermore, we will obtain a critical Sobolev type imbedding on these spaces.

Let Ω be a bounded Lipschitz domain in ℝ^N or Ω = ℝ^N Throughout this article, we assume that

s ∈ (0,1); p ∈ C ( Ω ¯ × Ω ¯ ) is uniformly continuous and symmetric such that

1 < p − := inf ( x , y ) ∈ Ω ¯ × Ω ¯ p ( x , y ) ≤ p + := sup ( x , y ) ∈ Ω ¯ × Ω ¯ p ( x , y ) < N s .

In the following, for brevity, we write p (x) instead of p (x, x) and with this notation, p ∈ C + ( Ω ¯ ) . Define

W s , p ( ⋅ , ⋅ ) ( Ω ) := u ∈ L p ( ⋅ ) ( Ω ) : ∫ Ω ∫ Ω | u ( x ) − u ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d x d y < + ∞

endowed with the norm

∥ u ∥ s , p , Ω := inf λ > 0 : M Ω u λ < 1 ,

where M Ω ( u ) := ∫ Ω u p ( x ) d x + ∫ Ω ∫ Ω | u ( x ) − u ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d x d y . Then, W s , p ( ⋅ , ⋅ ) ( Ω ) is a separable reflexive Banach space (see [11, 9, 8]). On W s , p ( ⋅ , ⋅ ) ( Ω ) , we also make use of the following norm

| u | s , p , Ω := | | u | | L p ( ⋅ ) ( Ω ) + [ u ] s , p , Ω ,

where

[ u ] s , p , Ω := inf λ > 0 : ∫ Ω ∫ Ω | u ( x ) − u ( y ) | p ( x , y ) λ p ( x , y ) | x − y | N + s p ( x , y ) d x d y < 1 .

Note that ∥ ⋅ ∥ s , p , Ω and | ⋅ | s , p , Ω are equivalent norms on W s , p ( ⋅ , ⋅ ) ( Ω ) with the relation

(3.1) 1 2 ∥ u ∥ s , p , Ω ≤ | u | s , p , Ω ≤ 2 ∥ u ∥ s , p , Ω , ∀ u ∈ W s , p ( ⋅ , ⋅ ) ( Ω ) .

In what follows, when Ω is understood, we just write ∥⋅∥s,p , | ⋅ | s , p and [ ⋅ ] s , p instead of ∥⋅∥s,p,Ω , |⋅|s,p,Ω and [ ⋅ ]s,p,Ω , respectively. We also denote the ball in ℝ^N centered at z with radius ε by B ε ( z ) and denote the Lebesgue measure of a set E⊂ℝN by |E| . For brevity, we write Bε and Bεc instead of Bε(0) and ℝN\Bε(0) , respectively.

Proposition 3.1

([10]) On W s , p ( ⋅ , ⋅ ) ( Ω ) it holds that

for u ∈ W s , p ( ⋅ , ⋅ ) ( Ω ) ∖ { 0 } , λ =∥ u ∥ s , p if and only if M Ω ( u λ ) = 1 ;
M Ω ( u ) > 1 ( = 1 ; < 1 ) if and only if ∥ u ∥ s , p > 1 ( = 1 ; < 1 ) , respectively;
if ∥ u ∥ s , p ≥ 1 , then ∥ u ∥ s , p p − ≤ M Ω ( u ) ≤∥ u ∥ s , p p + ;
if ∥u∥s,p<1 , then ∥ u ∥ s , p p + ≤ M Ω ( u ) ≤∥ u ∥ s , p p − .

Theorem 3.2

(Subcrtitical imbeddings, [10]) It holds that

W s , p ( ⋅ , ⋅ ) ( Ω ) ↪↪ L r ( ⋅ ) ( Ω ) , if Ω is a bounded Lipschitz domain and r ∈ C + ( Ω ¯ ) such that r ( x ) < N p ( x ) N − s p ( x ) =: p s ∗ ( x ) for all x ∈ Ω ¯ ;
W s , p ( ⋅ , ⋅ ) ( R N ) ↪ L r ( ⋅ ) ( R N ) for any uniformly continuous function r∈C+(ℝN) satisfying p(x)≤r(x) for all x∈ℝN and inf x ∈ R N ( p s ∗ ( x ) − r ( x ) ) > 0 ;
W s , p ( ⋅ , ⋅ ) ( R N ) ↪↪ L loc r ( ⋅ ) ( R N ) for any r∈C+(ℝN) satisfying r(x)<ps∗(x) for all x∈ℝN.

The next critical imbedding is our first main result.

Theorem 3.3

(Critical imbedding) Let Ω be a bounded Lipschitz domain in ℝ^N or Ω = ℝ^N. Let ( P 1 ) hold. Furthermore, let the variable exponent P satisfy the following log-Hölder type continuity condition

(3.2)infε>0 sup(x,y)∈Ω×Ω0<|x−y|<1/2|p(x,y)−pΩx,ε×Ωy,ε−|log1|x−y|<∞,

where Ωz,ε:=Bε(z)∩Ω for z∈Ω and ε>0 , and pΩx,ε×Ωy,ε−:=inf(x′,y′)∈Ωx,ε×Ωy,εp(x′,y′). Let q:Ω¯→ℝ be a function satisfying

( Q 1 ) q ∈ C + ( Ω ¯ ) such that for any x∈Ω , there exists ε=ε(x)>0 such that the following locally critical growth condition holds:

(3.3) sup y ∈ Ω x , ε q ( y ) ≤ N inf ( y , z ) ∈ Ω x , ε × Ω x , ε p ( y , z ) N − s inf ( y , z ) ∈ Ω x , ε × Ω x , ε p ( y , z ) .

In addition, when Ω = ℝ^N, q is uniformly continuous and p(x)<q(x) for all x ∈ ℝ^N.

Then, it holds that

(3.4) W s , p ( ⋅ , ⋅ ) ( Ω ) ↪ L q ( ⋅ ) ( Ω ) .

Proof

By the closed graph theorem, to prove (3.4) it suffices to show that W s , p ( ⋅ , ⋅ ) ( Ω ) ⊂ L q ( ⋅ ) ( Ω ) . Let u ∈ W s , p ( ⋅ , ⋅ ) ( Ω ) ∖ { 0 } be arbitrary and fixed. We will show that u ∈ L q ( ⋅ ) ( Ω ) , namely,

(3.5) ∫ Ω | u | q ( x ) d x < ∞ .

To this end, we first note that by (3.2), there exists constant ε0∈(0,1) such that

(3.6) sup ( x , y ) ∈ Ω × Ω 0 < | x − y | < 1 / 2 p ( x , y ) − p Ω x , ε 0 × Ω y , ε 0 − log ⁡ 1 | x − y | < C .

Here and in the remainder of the proof, C denotes a positive constant independent of u and may vary from line to line. We consider the following two cases.

Case 1: Ω is a bounded Lipschitz domain.

We cover Ω̅ by { B ε i ( x i ) } i = 1 m with xi∈Ω and εi∈(0,ε0) such that Ω i := B ε i ( x i ) ∩ Ω being Lipschitz domains and the locally critical growth condition (3.3) being satisfied for all i∈{1,⋯,m} . Fix i∈{1,⋯,m} and denote p i := inf ( y , z ) ∈ Ω i × Ω i ⁡ p ( y , z ) and q i := sup x ∈ Ω i ⁡ q ( x ) . By (3.3) and the choice of εi , we have

q i ≤ N p i N − s p i =: p s , i ∗ .

From this and Proposition 2.4, we have

∫ Ω i | u | q i d x ≤ C ∫ Ω i | u | p i d x + ∫ Ω i ∫ Ω i | u ( x ) − u ( y ) | p i | x − y | N + s p i d x d y q i p i ,

and hence,

(3.7) ∫ Ω i | u | q ( x ) d x − Ω i ≤ C Ω i + ∫ Ω i | u | p ( x ) d x + ∫ Ω i ∫ Ω i | u ( x ) − u ( y ) | p i | x − y | N + s p i d x d y q i p i .

On the other hand, we have

(3.8)∫Ωi∫Ωi|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dx dy=∫Ωi∫Ωi|u(x)−u(y)||x−y|2sp(x,y)1|x−y|N−spi1|x−y|−p(x,y)−pisdx dy.

Note that by (3.6), we have

(3.9) | x − y | − s p ( x , y ) − p i = e − s p ( x , y ) − p i log ⁡ | x − y | ≤ C , ∀ x , y ∈ Ω , x ≠ y .

Thus, (3.8) yields

∫Ωi∫Ωi|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dx dy≥C∫Ωi∫Ωi|u(x)−u(y)||x−y|2sp(x,y)1|x−y|N−spi dx dy≥C∫Ωi∫Ωi|u(x)−u(y)||x−y|2spi−11|x−y|N−spi dx dy

Hence,

(3.10)∫Ωi∫Ωi|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dx dy≥C∫Ωi∫Ωi|u(x)−u(y)|pi|x−y|N+spi dx dy−C∫Ωi∫Ωi1|x−y|N−spi dx dy.

We have

(3.11) ∫ Ω i ∫ Ω i 1 | x − y | N − s p i d x d y ≤ ∫ Ω d y ∫ B 2 d z | z | N − s p i = | Ω | N B 1 2 s p i s p i .

From (3.7), (3.10) and (3.11), we obtain

∫Ωi|u|q(x)dx≤C1+∫Ωi|u|p(x)dx+∫Ωi∫Ωi|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dx dyqipi≤C1+∫Ω|u|p(x)dx+∫Ω∫Ω|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dx dyg+p.

Summing up for i=1,⋯,m , we arrive at

∫ Ω | u | q ( x ) d x ≤ C 1 + ∫ Ω | u | p ( x ) d x + ∫ Ω ∫ Ω | u ( x ) − u ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d x d y q + p − < ∞ ,

and so (3.5) is claimed.

Case 2: Ω = R N .

Decompose R N by cubes { Q i } i ∈ N with sides of length ε∈(0,1) and parallel to coordinates axes. By the locally critical growth condition (3.3) and the uniform continuity of q we can choose ε>0 sufficiently small such that

(3.12) p i ≤ p i + ≤ q i − ≤ q i ≤ N p i N − s p i =: p s , i ∗ , ∀ i ∈ N ,

where

p i := inf ( y , z ) ∈ Q i × Q i p ( y , z ) , p i + := sup ( y , z ) ∈ Q i × Q i p ( y , z ) , q i − := inf x ∈ Q i q ( x ) , and q i := sup x ∈ Q i q ( x ) .

Set v=u∥u∥s,p . Thus, ∥v∥s,p=1 and hence, MℝN(v)=1 in view of Proposition 3.1. This yields MQi(v)≤1 for all i є ℕ and hence,

(3.13) ∥ v ∥ s , p , Q i ≤ 1 , ∀ i ∈ N .

We claim that

(3.14) | | v | | L q ( ⋅ ) ( Q i ) ≤ C ∥ v ∥ s , p , Q i , ∀ i ∈ N .

Here and in the remainder of the proof, C is a positive constant independent of v and i. In order to prove (3.14), we first prove that

(3.15) | | v | | s , p i , Q i ≤ C | | v | | s , p , Q i , ∀ i ∈ N .

Indeed, let i є ℕ and consider the measure 𝜇 on R N × R N such that

d μ ( x , y ) = d x d y | x − y | N − s p i .

As in (3.11) we have

(3.16) μ Q i × Q i ≤ Q i N B 1 2 s p i s p i < N B 1 2 s p + s p − , ∀ i ∈ N .

Set λ := [ v ] s , p , Q i and F ( x , y ) := | v ( x ) − v ( y ) | x − y | 2 s . Invoking Proposition 3.1 and (3.9) we estimate

1=∫Qi∫Qi|v(x)−v(y)|p(x,y)λp(x,y)|x−y|N+sp(x,y)dx dy=∫Qi∫Qi|v(x)−v(y)p(x,y)|λ|x−y2s1|x−y|−sp(x,y)−pidx dy|x−y|N−spi≥(C+1)−1∫Qi×QiF(x,y)λp(x,y)dμ(x,y)≥∫Qi×Qi(C+1)−p(x,y)piF(x,y)λp(x,y)dμ(x,y)=∫Qi×QiF(x,y)(C+1)1piλp(x,y)dμ(x,y).

Thus,

∥ F ∥ L μ p ( ⋅ , ⋅ ) ( Q i × Q i ) ≤ ( C + 1 ) 1 p i λ = ( C + 1 ) 1 p i [ v ] s , p , Q i .

Meanwhile, invoking Proposition 2.1 we have

∥ F ∥ L μ p i ( Q i × Q i ) ≤ 2 ( 1 + μ ( Q i × Q i ) ) ∥ F ∥ L μ p ( ⋅ , ⋅ ) ( Q i × Q i ) .

Combining the last two inequalities and (3.16) we obtain

(3.17) ∥ F ∥ L μ p i ( Q i × Q i ) ≤ C [ v ] s , p , Q i .

Noting

we deduce from (3.17) that

(3.18) [ v ] s , p i , Q i ≤ C [ v ] s , p , Q i .

Combining (3.18) with the following estimate:

∥ v ∥ L p i Q i ≤ 2 1 + Q i ∥ v ∥ L p ( ⋅ ) Q i ≤ 4 ∥ v ∥ L p ( ⋅ ) Q i

(see Proposition 2.1) and the relation (3.1), we obtain (3.15).

As in [10, Proof of Theorem 3.5], we can obtain an extension v ~ ∈ W s , p i R N with compact support in ℝ^N such that v ~ = v on Q_i, and

∥ v ~ ∥ L p s , i ∗ ( R N ) ≤ C ∥ v ∥ s , p i , Q i .

This and (3.12) yield

(3.19) ∥ v ∥ L q i Q i ≤ C ∥ v ∥ s , p i , Q i .

Note that by Proposition 2.1 again,

(3.20) ∥ v ∥ L q ( ⋅ ) Q i ≤ 2 1 + Q i ∥ v ∥ L q i Q i ≤ 4 ∥ v ∥ L q i Q i .

From (3.15), (3.19), and (3.20) we obtain (3.14). Now, for each i ∈ N , if ∥ v ∥ L q ( ⋅ ) Q i ≥ 1 , then by (3.13), (3.14) and Proposition 3.1 we have

∫ Q i | v | q ( x ) d x ≤ ∥ v ∥ L q ( ⋅ ) Q i q i ≤ C ∥ v ∥ s , p , Q i q i ≤ C ∫ Q i | v | p ( x ) d x + ∫ Q i ∫ Q i | v ( x ) − v ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d x d y q i p i + ≤ C ∫ Q i | v | p ( x ) d x + ∫ Q i ∫ Q i | v ( x ) − v ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d x d y .

Similarly, if ∥ v ∥ L q ( ⋅ ) Q i ≤ 1 , , then

∫ Q i | v | q ( x ) d x ≤ ∥ v ∥ L q ( ⋅ ) Q i q ¯ i ≤ C ∥ v ∥ s , p , Q i q ¯ i ≤ C ∫ Q i | v | p ( x ) d x + ∫ Q i ∫ Q i | v ( x ) − v ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d x d y q ¯ i p i + ≤ C ∫ Q i | v | p ( x ) d x + ∫ Q i ∫ Q i | v ( x ) − v ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d x d y .

So in any case,

∫ Q i | v | q ( x ) d x ≤ C ∫ Q i | v | p ( x ) d x + ∫ Q i ∫ Q i | v ( x ) − v ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d x d y .

Summing up for i ∈ N , , we obtain

∫ R N | v | q ( x ) d x ≤ C ∫ R N | v | p ( x ) d x + ∫ R N ∫ R N | v ( x ) − v ( y ) | p ( x , y ) | x − y | N + sp ⁡ ( x , y ) d x d y ,

which implies (3.5) with Ω = ℝ^N The proof is complete.

We conclude this section with a compact imbedding from W S , p ( ⋅ , ⋅ ) R N into the weighted Lebesgue spaces with variable exponents.

Theorem 3.4

Assume that (𝒫₁), (Q₁), and the log-Hölder continuity condition (3.2) hold. Let W ∈ L + q ( ⋅ ) q ( ⋅ ) − r ( ⋅ ) R N for some r ∈ C + R N such that inf x ∈ R N [ q ( x ) − r ( x ) ] > 0 . Then, it holds that

W S , p ( ⋅ , ⋅ ) R N ↪↪ L r ( ⋅ ) w , R N .

A proof of Theorem 3.4 can be obtained in a similar fashion to that of [27, Lemma 4.1] and we omit it.

4 The concentration-compactness principles for W s , p ( ⋅ , ⋅ ) ( R N )

In this section we establish two Lions type concentration-compactness principles for the spaces W s , p ( ⋅ , ⋅ ) ( R N ) .

4.1 Statements of the concentration-compactness principles

Let 𝛭(ℝ^N) be the space of all signed finite Radon measures on ℝ^N endowed with the total variation norm. Note that we may identify 𝛭(ℝ^N) with the dual of C 0 R N , , the completion of all continuous functions u : R N → R whose support is compact relative to the supremum norm ∥ ⋅ ∥ ∞ (see, e.g., [32, Section 1.3.3]).

In the rest of this paper, we always assume that the variable exponents p and q satisfy the following assumptions.

(𝒫₂) p:ℝN×ℝN→ℝ is uniformly continuous and symmetric such that

1 < p − := inf ( x , y ) ∈ R N × R N p ( x , y ) ≤ sup ( x , y ) ∈ R N × R N p ( x , y ) =: p ¯ < N s ;

there exists ε0∈(0,12) such that p(x,y)=p¯ for all x,y∈ℝN satisfying |x−y|<ε0 and supy∈ℝNp(x,y)=p¯ for all x∈ℝN; and |{x∈ℝN: p∗(x)=/p¯}|<∞, where p*(x):=infy∈ℝNp(x,y) for x∈ℝN .

(𝓠₂) q:ℝN→ℝ is uniformly continuous such that p∗(x)≤q(x)≤p¯s∗:=Np¯N−sp¯ for all x∈ℝN and C := { x ∈ R N : q ( x ) = p ¯ s ⋆ ≠ ∅ .

It is clear that if p satisfies (𝒫₂ ), then p(x,x)=p¯ for all x∈ℝN and p satisfies (𝒫₁) and (3.2). Hence, by Theorem 3.3, we have

(4.1) W s , p ( ⋅ , ⋅ ) R N ↪ L p ¯ s ⋆ R N .

On the other hand, by (𝒫₂) we have that for any u∈Lp¯(ℝN) ,

∫ R N | u ( x ) | p ∗ ( x ) d x = ∫ { p ∗ ( x ) = p ¯ } | u ( x ) | p ∗ ( x ) d x + ∫ { p ∗ ( x ) ≠ p ¯ } | u ( x ) | p ∗ ( x ) d x ≤ ∫ p ∗ ( x ) = p ¯ | u ( x ) | p ¯ d x + ∫ { p ∗ ( x ) ≠ p ¯ } 1 + | u ( x ) | p ¯ d x = χ ∈ R N : p ⋆ ( x ) ≠ p ¯ + ∫ R N | u ( x ) | p ¯ d x < ∞ .

Hence, L p ¯ ( R N ) ⊂ L p ∗ ( ⋅ ) ( R N ) . From this and (4.1) we obtain

(4.2) W s , p ( ⋅ , ⋅ ) R N ↪ L t ( ⋅ ) R N

for any t∈C(ℝN) satisfying p∗(x)≤t(x)≤p¯s∗ for all x∈ℝN. In particular, (𝓠₂) yields

(4.3) S q := inf u ∈ W s , p ( ⋅ , ⋅ ) R N ∖ { 0 } ∥ u ∥ s , p ∥ u ∥ L q ( ⋅ ) R N > 0.

Our main results in this sections are the following CCPs for W s , p ( ⋅ , ⋅ ) ( R N ) .

Theorem 4.1

Assume that (𝒫₂) and (𝓠₂) hold. Let {u_n} be a bounded sequence in W s , p ( ⋅ , ⋅ ) ( R N ) such that

(4.4) u n ⇀ u in W s , p ( ⋅ , ⋅ ) ( R N ) ,

(4.5) u n p ¯ + ∫ R N u n ( x ) − u n ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y ⇀ ⋆ μ in M R N ,

(4.6)|un|q(x) ⇀∗ ν in M(ℝN).

Then, there exist sets { μ i } i ∈ I ⊂ ( 0 , ∞ ) , { ν i } i ∈ I ⊂ ( 0 , ∞ ) and { x i } i ∈ I ⊂ C , where I is an at most countable index set, such that

(4.7) μ ≥ | u | p ¯ + ∫ R N | u ( x ) − u ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d y + ∑ i ∈ I μ i δ x i ,

(4.8) ν = | u | q ( x ) + ∑ i ∈ I ν i δ x i ,

(4.9) S q v i 1 p ¯ s ∗ ≤ μ i 1 p ¯ , ∀ i ∈ I .

For possible loss of mass at infinity, we have the following.

Theorem 4.2

Assume that (𝒫₂) and (𝓠₂) hold. Let {u_n} be a sequence in W s , p ( ⋅ , ⋅ ) ( R N ) as in Theorem 4.1. Set

(4.10) v ∞ := lim R → ∞ lim sup n → ∞ ∫ B R c u n q ( x ) d x ,

(4.11) μ ∞ := lim R → ∞ lim sup n → ∞ ∫ B R c u n p ¯ + ∫ R N u n ( x ) − u n ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x .

Then

(4.12) lim sup n → ∞ ∫ R N u n q ( x ) d x = v R N + v ∞ ,

(4.13) lim sup n → ∞ ∫ R N u n p ¯ + ∫ R N u n ( x ) − u n ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x = μ R N + μ ∞ .

Assume in addition that

(ε_∞) There exist lim|x|,|y|→∞p(x,y)=p¯ and lim|x|→∞q(x)=q∞ for p̅ given by (𝒫₂) and some q ∞ ∈ ( 1 , ∞ ) .

Then

(4.14) S q ν ∞ 1 q ∞ ≤ μ ∞ 1 p ¯ .

The following example provides a nonconstant exponent p that fulfills the conditions in Theorems 4.1 and 4.2.

Example 4.3

Let p(x,y)=p¯−ξ(|x−y|)φ(x,y) , where ξ ∈ C ∞ ( R ) such that 0 ≤ ξ ( t ) ≤ 1 for all t ∈ R , ξ ( t ) = 0 for t ≤ ε 0 and ξ ( t ) = 1 for t ≥ 1 ; φ ∈ C c ∞ ( R N × R N ) , φ ( x , y ) = φ ( y , x ) and 0 ≤ φ ( x , y ) < p ¯ − 1 for all ( x , y ) ∈ R N × R N . Here ε₀ and p̅ are as in (𝒫₂).

4.2 Auxiliary lemmas and proofs of the concentration-compactness principles

The following auxiliary lemmas are useful to prove Theorems 4.1 and 4.2.

Lemma 4.4

Let x 0 ∈ R N be fixed and let ψ ∈ C ∞ ( R N ) be such that 0 ≤ ψ ≤ 1 , ψ ≡ 1 on B₁, supp ⁡ ( ψ ) ⊂ B 2 and | | ∇ ψ | | ∞ ≤ 2 . For ρ > 0 , define ψ ρ ( x ) := ψ ( x − x 0 ρ ) for x ∈ ℝ^N. Let (𝒫₂ ) hold and let {u_n} be as in Theorem 4.1. Then, we have

(4.15)lim supρ→0+lim supn→∞∫RN∫RNun(x)p(x,y)ψρ(x)−ψρ(y)p(x,y)|x−y|N+sp(x,y)dy dx=0.

Lemma 4.5

Let ϕ ∈ C ∞ ( R N ) be such that 0 ≤ ϕ ≤ 1 , ϕ ≡ 0 on B₁, ϕ ≡ 1 on R N ∖ B 2 , and | | ∇ ϕ | | ∞ ≤ 2. For R > 0 , define ϕ R ( x ) := ϕ ( x R ) for x ∈ ℝ^N Let (𝒫₂ ) hold and let {u_n} be as in Theorem 4.1. Then, we have

(4.16) lim R → ∞ lim sup n → ∞ ∫ R N ∫ R N u n ( x ) p ( x , y ) ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + sp ⁡ ( x , y ) d y d x = 0.

Proof of Lemma 4.4. Set

J ( n , ρ ) = ∫ R N ∫ R N u n ( x ) p ( x , y ) ψ ρ ( x ) − ψ ρ ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x .

Let K > 4 be arbitrary and fixed and let ρ ∈ ( 0 , ε 0 2 K ) . Clearly,

R N × R N = ( R N ∖ B 2 ρ ( x 0 ) ) × ( R N ∖ B 2 ρ ( x 0 ) ) ∪ B K ρ ( x 0 ) × R N ∪ ( R N ∖ B K ρ ( x 0 ) ) × B 2 ρ ( x 0 ) .

From this and the fact that | ψ ρ ( x ) − ψ ρ ( y ) | = 0 on ( R N ∖ B 2 ρ ( x 0 ) ) × ( R N ∖ B 2 ρ ( x 0 ) ) , we have

(4.17) J ( n , ρ ) = ∫ B K ρ x 0 ∫ R N u n ( x ) p ( x , y ) ψ ρ ( x ) − ψ ρ ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x + ∫ R N ∖ B K ρ x 0 ∫ B 2 ρ x 0 u n ( x ) p ( x , y ) ψ ρ ( x ) − ψ ρ ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x =: J 1 ( n , ρ ) + J 2 ( n , ρ ) .

We first estimate J 1 ( n , ρ ) . Decompose

(4.18) J 1 ( n , ρ ) = ∫ B K ρ x 0 ∫ { | x − y | ≤ ρ } u n ( x ) p ( x , y ) ψ ρ ( x ) − ψ ρ ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x + ∫ B K ρ x 0 ∫ ρ < | x − y | < ε 0 u n ( x ) p ( x , y ) ψ ρ ( x ) − ψ ρ ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x + ∫ B K ρ x 0 ∫ | x − y | ≥ ε 0 u n ( x ) p ( x , y ) ψ ρ ( x ) − ψ ρ ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x =: ∑ i = 1 3 J 1 ( i ) ( n , ρ ) .

By (𝒫₂) and the choice of ψρ , we have p(x,y)=p¯ and |ψρ(x)−ψρ(y)|≤2ρ|x−y| on {|x−y|<ρ} . Thus, we obtain

J 1 ( 1 ) ( n , ρ ) ≤ 2 ρ p ¯ ∫ B K ρ x 0 u n ( x ) p ¯ ∫ { | x − y | ≤ ρ } | x − y | − N + ( 1 − s ) p ¯ d y d x .

Hence,

(4.19) J 1 ( 1 ) ( n , ρ ) ≤ 2 p ¯ N B 1 ( 1 − s ) p ¯ ρ − s p ¯ ∫ B K ρ x 0 u n ( x ) p ¯ d x .

By (4.4), we have that un→u in Lp¯(BKρ(x0)) in view of and Theorem 3.2 (iii). From this fact and (4.19) we obtain

(4.20) lim sup n → ∞ J 1 ( 1 ) ( n , ρ ) ≤ 2 p ¯ N B 1 ( 1 − s ) p ¯ ρ − s p ¯ ∫ B K ρ x 0 | u ( x ) | p ¯ d x .

Using the Hölder inequality we have

(4.21) ∫ B K ρ x 0 | u ( x ) | p ¯ d x ≤ B 1 s p ¯ N K s p ¯ ρ s p ¯ ∫ B K ρ x 0 | u ( x ) | p ¯ s ∗ d x p ¯ p ¯ s ∗

From (4.20), (4.21) and the fact that u∈Lp¯s∗(ℝN) (see (4.1)) we arrive at

(4.22) lim sup ρ → 0 + lim sup n → ∞ J 1 ( 1 ) ( n , ρ ) = 0.

On the other hand, from p ( x , y ) = p ¯ and | ψ ρ ( x ) − ψ ρ ( y ) | p ( x , y ) ≤ 1 on { ρ < | x − y | < ε 0 } we have

J 1 ( 2 ) ( n , ρ ) ≤ ∫ B K ρ x 0 u n ( x ) p ¯ ∫ ρ < | x − y | < ε 0 | x − y | − N − s p ¯ d y d x .

That is,

(4.23) J 1 ( 2 ) ( n , ρ ) ≤ N B 1 s p ¯ ∫ B K ρ x 0 u n ( x ) p ¯ ρ − s p ¯ − ε 0 − s p ¯ d x .

Arguing as that obtained (4.20) we deduce from (4.23) that

(4.24) lim sup n → ∞ J 1 ( 2 ) ( n , ρ ) ≤ N B 1 s p ¯ ∫ B K ρ x 0 | u ( x ) | p ¯ ρ − s p ¯ − ε 0 − s p ¯ d x .

Then, using (4.21) and the fact that u ∈ L p ¯ s ∗ ( R N ) again we obtain from (4.24) that

(4.25) lim sup ρ → 0 + lim sup n → ∞ J 1 ( 2 ) ( n , ρ ) = 0.

In order to estimate J 1 ( 3 ) ( n , ρ ) , using the following estimations on { | x − y | ≥ ε 0 } :

| u n ( x ) | p ( x , y ) ≤ 1 + | u n ( x ) | p ¯

and

| ψ ρ ( x ) − ψ ρ ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) ≤ 1 | x − y | N + s p ( x , y ) ≤ 1 | x − y | N + s p − + 1 | x − y | N + s p ¯ ,

we first estimate

J 1 ( 3 ) ( n , ρ ) ≤ ∫ B K ρ x 0 1 + u n ( x ) p ¯ ∫ | x − y | ≥ ε 0 1 | x − y | N + s p − + 1 | x − y | N + s p ¯ d y d x ≤ 2 N B 1 ε 0 − s p ¯ s p − ∫ B K ρ x 0 1 + u n ( x ) p ¯ d x .

Then, arguing as before we obtain

(4.26) lim sup ρ → 0 + lim sup n → ∞ J 1 ( 3 ) ( n , ρ ) = 0.

Utilizing (4.22), (4.25) and (4.26), we infer from (4.18) that

(4.27) lim sup ρ → 0 + lim sup n → ∞ J 1 ( n , ρ ) = 0.

Next, we estimate J 2 ( n , ρ ) . Note that

(4.28) J 2 ( n , ρ ) ≤ ∑ t ∈ { p ∗ , p ¯ } ∫ R N ∖ B K ρ x 0 ∫ B 2 ρ x 0 u n ( x ) t ( x ) ψ ρ ( x ) − ψ ρ ( y ) t ( x ) | x − y | N + s t ( x ) d y d x =: ∑ t ∈ p ∗ , p ¯ J 2 ( n , ρ , t ) .

Let t ∈ { p ∗ , p ¯ } . Using the fact that

| x − y | ≥ | x − x 0 | − | y − x 0 | ≥ | x − x 0 | − 2 ρ ≥ 1 2 | x − x 0 | , ∀ ( x , y ) ∈ ( R N ∖ B K ρ ( x 0 ) ) × B 2 ρ ( x 0 ) ,

we have

J 2 ( n , ρ , t ) ≤ 2 N + s p ¯ ∫ R N ∖ B K ρ x 0 u n ( x ) t ( x ) x − χ 0 N + s t ( x ) ∫ B 2 p x 0 d y d x .

That is,

J 2 ( n , ρ , t ) ≤ 2 2 N + s p ¯ B 1 ∫ R N ∖ B K ρ x 0 u n ( x ) t ( x ) x − x 0 N + s t ( x ) ρ N d x .

Invoking Proposition 2.3 again and using the boundedness of {u_n} in L p ¯ s ∗ ( R N ) , we deduce from the last inequality that

(4.29) J 2 ( n , ρ , t ) ≤ C 1 u n t ( x ) L p ¯ s ∗ t ( ⋅ ) R N ∖ B K p x 0 x − x 0 − N − s t ( x ) ρ N p ¯ s ∗ L p ¯ s ∗ − t ( ⋅ ) R N ∖ B K ρ x 0 ≤ C 2 max ∫ R N ∖ B K ρ x 0 X − X 0 − ( N + s t ( x ) ) p → s ∗ p → s ∗ − t ( x ) ρ N p ¯ s ∗ p ¯ s ∗ − t ( x ) d x p ¯ s ∗ − t p ¯ s ∗ + , ∫ R N ∖ B K ρ x 0 x − X 0 − ( N + st ⁡ ( x ) ) p → s ∗ P ¯ S ∗ − t ( x ) ρ N p s ∗ p ¯ s ∗ − t ( x ) p ¯ s ∗ − t p ¯ s ∗ − .

Here and in the remainder of the proof C_i (i є ℕ) is a positive constant independent of n,ρ and K. By changing variable x = x 0 + ρ z we have

(4.30) ∫ R N ∖ B K ρ x 0 χ − χ 0 − ( N + s t ( x ) ) p ¯ s ⋆ p ¯ s ∗ − t ( x ) ρ N p ¯ s ⋆ p ¯ s ∗ − t ( x ) d x = ∫ { | z | ≥ K } | z | − N + s t x 0 + ρ z p ¯ s ⋆ p ¯ s ∗ − t x 0 + ρ z ρ N − s t x 0 + ρ z p ¯ s ⋆ p ¯ s ∗ − t x 0 + ρ z d z .

Note that for any x ∈ ℝ^N, it holds that N − s t ( x ) p ¯ s ∗ p ¯ s ∗ − t ( x ) ≥ 0 due to t ( x ) ≤ p ¯ and ( N + s t ( x ) ) p ¯ s ∗ p ¯ s ⋆ − t ( x ) = N + t ( x ) N + s p ¯ s ∗ p ¯ s ∗ − t ( x ) > N + α , where α := N + p ¯ s ∗ p ¯ s ∗ − 1 > 0 . Plugging this into (4.30) we obtain

∫ R N ∖ B K ρ x 0 χ − χ 0 − ( N + s t ( x ) ) p ¯ s ⋆ p ¯ s ∗ − t ( x ) ρ N p → s ⋆ p ¯ s ∗ − t ( x ) d x ≤ ∫ { | z | ≥ K } | z | − N − α d z = N B 1 α K − α .

Combining this with (4.28) and (4.29) we derive

J 2 ( n , ρ ) ≤ C 3 K − α p ¯ s ∗ − p ¯ p ¯ s ∗

for all n ϵ N and all ρ ∈ ( 0 , ε 0 2 K ) . Thus,

lim sup ρ → 0 + lim sup n → ∞ J 2 ( n , ρ ) ≤ C 3 K − α p ¯ − ∗ − p ¯ p → s .

Since K > 4 was chosen arbitrarily, the last inequality yields

(4.31) lim sup ρ → 0 + lim sup n → ∞ J 2 ( n , ρ ) = 0.

From (4.17), (4.27), and (4.31), we obtain (4.16) and the proof is complete.

Proof of Lemma 4.5. Let R>2 and decompose

(4.32) ∫ R N ∫ R N u n ( x ) p ( x , y ) ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x = ∫ B R c ∫ R N u n ( x ) p ( x , y ) ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x + ∫ B R ∫ R N u n ( x ) p ( x , y ) ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x =: I 1 ( n , R ) + I 2 ( n , R ) .

First, we estimate I 1 ( n , R ) . By rearranging

I 1 ( n , R ) = ∫ B R c ∫ R N u n ( x ) ∥ ϕ R ( x ) − ϕ R ( y ) | x − y | s p ( x , y ) 1 | x − y | N d y d x ,

we easily get

(4.33) I 1 ( n , R ) ≤ ∫ B R c ∫ R N | u n ( x ) | p ¯ | ϕ R ( x ) − ϕ R ( y ) | p ¯ | x − y | s p ¯ + | u n ( x ) | p ∗ ( x ) | ϕ R ( x ) − ϕ R ( y ) | p ∗ ( x ) | x − y | s p ∗ ( x ) d y d x | x − y | N = ∫ B R c ∫ R N | u n ( x ) | p ¯ | ϕ R ( x ) − ϕ R ( y ) | p ¯ | x − y | N + s p ¯ d y d x + ∫ B R c ∫ R N | u n ( x ) | p ∗ ( x ) | ϕ R ( x ) − ϕ R ( y ) | p ∗ ( x ) | x − y | N + s p ∗ ( x ) d y d x =: I 1 ( n , R , p ¯ ) + I 1 ( n , R , p ∗ ) .

By (4.2) and the boundedness of {u_n} in W s , p ( ⋅ , ⋅ ) ( R N ) , we can find M>0 such that

(4.34) max t ∈ { p ¯ , p ⋅ } sup n ∈ N ∫ R N u n ( x ) t ( x ) d x ≤ M .

Let t ∈ { p ¯ , p ∗ } . We have

(4.35) I 1 ( n , R , t ) = ∫ B R c ∫ R N u n ( x ) t ( x ) ϕ R ( x ) − ϕ R ( y ) t ( x ) | x − y | N + s t ( x ) d y d x = ∫ B R c u n ( x ) t ( x ) ∫ { | x − y | ≥ R } ϕ R ( x ) − ϕ R ( y ) t ( x ) | x − y | N + st ⁡ ( x ) d y d x + ∫ B R c u n ( x ) t ( x ) ∫ { | x − y | ≤ R } ϕ R ( x ) − ϕ R ( y ) t ( x ) | x − y | N + st ⁡ ( x ) d y d x = : I 1 ( 1 ) ( n , R , t ) + I 1 ( 2 ) ( n , R , t ) .

We have

I 1 ( 1 ) ( n , R , t ) ≤ ∫ B R c u n ( x ) t ( x ) ∫ B R c d z | z | N + s t ( x ) d x = N B 1 ∫ B R c u n ( x ) t ( x ) s t ( x ) R s t ( x ) d x ≤ N B 1 s p − R s p − ∫ B R c u n ( x ) t ( x ) d x .

Combining this with (4.34) gives

(4.36) sup n ∈ N I 1 ( 1 ) ( n , R , t ) ≤ N B 1 M s p − R − s p − .

On the other hand, using estimation:

(4.37) ϕ R ( x ) − ϕ R ( y ) t ( x ) | x − y | N + s t ( x ) ≤ | x − y | − N + ( 1 − s ) t ( x ) ∇ ϕ R ∞ t ( x ) ≤ | x − y | − N + ( 1 − s ) t ( x ) 2 R t ( x ) , ∀ ( x , y ) ∈ R N × R N , x ≠ y ,

we have

I 1 ( 2 ) ( n , R , t ) ≤ ∫ B R c u n ( x ) t ( x ) ∫ { | x − y | ≤ R } 2 R t ( x ) | x − y | − N + ( 1 − s ) t ( x ) d y d x = ∫ B R c u n ( x ) t ( x ) 2 R t ( x ) N B 1 R ( 1 − s ) t ( x ) ( 1 − s ) t ( x ) d x ≤ 2 p ¯ N B 1 1 − s R − s p − ∫ B R c u n ( x ) t ( x ) d x .

Combining this and (4.34) yields

(4.38) sup n ∈ N I 1 ( 2 ) ( n , R , t ) ≤ 2 p ¯ N B 1 M 1 − s R − s p − .

From (4.33), (4.35), (4.36) and (4.38) we obtain

sup n ∈ N I 1 ( n , R ) ≤ 2 p ¯ + 1 N B 1 M s ( 1 − s ) R − s p −

and hence,

(4.39) lim R → ∞ lim sup n → ∞ I 1 ( n , R ) = 0

Next, we estimate I 2 ( n , R ) . Fix σ ∈ ( 0 , 1 / 2 ) and decompose

(4.40) I 2 ( n , R ) = ∫ B R ∖ B σ R ∫ | x − y | ≤ R 2 u n ( x ) p ( x , y ) ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + sp ⁡ ( x , y ) d y d x + ∫ B R ∖ B σ R ∫ | x − y | > R 2 u n ( x ) p ( x , y ) ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + sp ⁡ ( x , y ) d y d x + ∫ B σ R ∫ R N u n ( x ) p ( x , y ) ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + sp ⁡ ( x , y ) d y d x =: I 2 ( 1 ) ( n , R , σ ) + I 2 ( 2 ) ( n , R , σ ) + I 2 ( 3 ) ( n , R , σ ) .

Note that as in (4.37), we have

ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + s p ( x , y ) ≤ | x − y | − N | | x − y 1 − s 2 R p ( x , y ) ≤ 2 p ¯ ∑ t ∈ p ¯ , p − R − t | x − y | − N + ( 1 − s ) t , ∀ ( x , y ) ∈ R N × R N , x ≠ y .

Thus, for x ∈ B R ∖ B σ R we have

∫ | x − y | ≤ R 2 ϕ R ( χ ) − ϕ R ( y ) p ( x , y ) | χ − y | N + s p ( x , y ) d y ≤ 2 p ¯ ∑ t ∈ p ¯ , p − ∫ | x − y | ≤ R 2 R − t | χ − y | − N + ( 1 − s ) t d y ≤ 2 2 p ¯ N B 1 ∑ t ∈ p ¯ , p − R − s t ( 1 − s ) t .

From this and (4.34) we obtain

(4.41) I 2 ( 1 ) ( n , R , σ ) ≤ ∫ B R ∖ B σ R u n ( x ) p ¯ + u n ( x ) p ∗ ( x ) ∫ | x − y | ≤ R 2 ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x ≤ 2 1 + 2 p ¯ N B 1 ( 1 − s ) p − R − s p − ∫ B R ∖ B σ R u n ( x ) p ¯ + u n ( x ) p ∗ ( x ) d x ≤ 2 2 + 2 p ¯ N B 1 M ( 1 − s ) p − R − s p − , ∀ n ∈ N .

Using (4.34) again, we have

(4.42) I 2 ( 2 ) ( n , R , σ ) ≤ ∫ B R ∖ B σ R u n ( x ) p ¯ + u n ( x ) p ∗ ( x ) ∫ | x − y | > R 2 d y | x − y | N + s p ( x , y ) d x ≤ ∫ B R ∖ B σ R u n ( x ) p ¯ + u n ( x ) p ∗ ( x ) ∫ | z | > R 2 d z | z | N + s p − d x ≤ 2 1 + s p − N B 1 M s p − R − s p − , ∀ n ∈ N .

Finally, to estimate I 2 ( 3 ) ( n , R , σ ) we first note that ϕ R ( x ) − ϕ R ( y ) = 0 for all ( x , y ) ∈ B σ R × B R and | x − y | ≥ ( 1 − σ ) R for all ( x , y ) ∈ B σ R × B R c . Using these facts and invoking (4.34) again, we have

(4.43) I 2 ( 3 ) ( n , R , σ ) = ∫ B σ R ∫ B R c u n ( x ) p ( x , y ) ϕ R ( x ) − ϕ R ( y ) p ( x , y ) | x − y | N + s p ( x , y ) d y d x ≤ ∫ B σ R u n ( x ) p ¯ + u n ( x ) p ∗ ( x ) ∫ B R c d y | x − y | N + s p ( x , y ) d x ≤ ∫ B σ R u n ( x ) p ¯ + u n ( x ) p ∗ ( x ) ∫ { | z | ≥ ( 1 − σ ) R } d z | z | N + s p − d x ≤ 2 N B 1 ( 1 − σ ) − s p − M s p − R − s p − .

Making use of (4.41)-(4.43), we deduce from (4.40) that

(4.44)limR→∞limsupn→∞ I2(n,R)=0.

Finally, (4.15) follows from (4.32), (4.39) and (4.44). The proof is complete.□

We now prove the first concentration-compactness principle.

Proof of Theorem 4.1. Let v_n = u_n - u. Then,

(4.45)vn ⇀ 0 in Ws,p(⋅,⋅)(ℝN).

Invoking Theorem 3.2, we deduce from (4.45) that

(4.46)vn → 0 in Llocr(⋅)(ℝN)

for any r∈C+(ℝN) satisfying r(x)<p¯s∗ for all x∈ℝN . Hence, up to a subsequence we have

(4.47)vn(x) → 0 fora.e. x∈ℝN.

Using (4.6), (4.45), (4.47) and arguing as in [27], we have

(4.48)|vn|q(x)⇀∗ν−|u|q(x)=:ν¯inM(RN).

Obviously, vnp¯+∫RN∣vn(x)−vn(y)pp(x,y)|x−y|N+sp(x,y)dy is bounded in L1(RN) . So up to a subsequence, we have

(4.49)vnp¯+∫RNvn(x)−vn(y)p(x,y)|x−y|N+sp(x,y)dy⇀⋆μ¯ in MRN

for some nonnegative finite Radon measure μ̅ on ℝ^N. Let ϕ∈Cc∞(RN) and let R > 2 be such that

(4.50)supp(ϕ)⊂BRandd:=dist(BRc,supp(ϕ))≥1+R2.

By (4.3), we have

(4.51)Sq||ϕvn||Lq(⋅)(RN)≤||ϕvn||s,p.

Set ν¯n:=|vn|q(x), μ¯n:=vn(x)p¯+∫RNvn(x)−vn(y)p(x,y)|x−y|N+sp(x,y)dy , and λn:=∥ϕvn∥s,p . Let ε>0 be arbitrary and fixed. Then, there exists C(ε)∈(2,∞) such that

(4.52)|a+b|p(x,y)≤(1+ε)|a|p(x,y)+C(ε)|b|p(x,y),∀a,b∈R,∀x,y∈RN.

Invoking Proposition 3.1 and (4.52) we have

Set

In:=∫RN∫RNVn(y)p(x,y)|ϕ(x)−ϕ(y)|p(x,y)λnp(x,y)|x−y|N+sp⁡(x,y)dy dx.

Then, invoking Proposition 3.1 again we deduce from (4.53) that

(4.54)1≤(1+ε)(||ϕ||∞p¯+1)min{λnp¯,λnp−}1+∥vn∥s,pp¯+C(ε)In.

By the symmetry of P we also have

In=∫RN∫RNvn(x)p(x,y)|ϕ(x)−ϕ(y)|p(x,y)λnp(x,y)|x−y|N+sp(x,y)dy dx.

Thus, by the facts that supp (Φ) ⊂ B_R and λnp(x,y)≥min{λnp¯,λnp−} for all x,y∈RN ,

(4.55)In≤1minλnp¯,λnp−∫BRcvn(x)p∗(x)+vn(x)p¯∫BR|ϕ(x)−ϕ(y)|p(x,y)|x−y|N+sp(x,y)dy dx+∫BRvn(x)p+(x)+vn(x)p¯∫BRc|ϕ(x)−ϕ(y)|p(x,y)|x−y|N+sp(x,y)dy dx+∫BRvn(x)p⋅(x)+vn(x)p¯∫BR|ϕ(x)−ϕ(y)|p(x,y)|x−y|N+sp(x,y)dy dx.

We estimate each integral in the right-hand side of (4.55) as follows. Arguing as that obtained (4.34) we have

(4.56)maxt∈p¯,p⋆supn∈N∫RNvn(x)t(x)dx≤C1.

Here and in the rest of the proof, C_i (i є ℕ) denotes a positive constant independent of n and R while C_i (i є ℕ) denotes a positive constant independent of n. Let t∈{p¯,p∗} . Using (4.50) and (4.56), we have

(4.57)∫BRcvn(x)t(x)∫BR|ϕ(x)−ϕ(y)|p(x,y)|x−y|N+sp(x,y)dydx=∫BRcvn(x)t(x)∫supp⁡(ϕ)|ϕ(y)|p(x,y)|x−y|N+sp(x,y)dydx≤1+∥ϕ∥∞p¯∫BRcvn(x)t(x)∫supp⁡(ϕ)dyR2N+sp(x,y)dx≤1+∥ϕ∥∞p¯R2N+sp−BR∫RNvn(x)t(x)dx≤C2Rsp.

Before estimating the remaining integrals, we note that by (4.50) again,

(4.58)∫BRc|ϕ(x)−ϕ(y)|p(x,y)|x−y|N+sp(x,y)dy=∫BRc|ϕ(x)|p(x,y)|x−y|N+sp(x,y)dy≤1+∥ϕ∞p¯∫{|z|≥1}dz|z|N+sp−=1+∥ϕ∞p¯N|B1|sp−,∀x∈BR.

Using (4.58), we have

(4.59)∫BR|vn(x)|t(x)(∫BRc|ϕ(x)−ϕ(y)|p(x,y)|x−y|N+sp(x,y)dy)dx≤C3∫BR|vn(x)|t(x)dx.

To estimate the last integral in the right-hand side of (4.55) we notice that for x∈BR,

∫Bg|ϕ(χ)−ϕ(y)|p(x,y)|x−y|N+sp(x,y)dy≤1+∥∇ϕ∥∞p¯∫BRdy|x−y|N+(s−1)p(x,y)≤1+∥∇ϕ∥∞p¯∫BR1+1|x−y|N+(s−1)pdy≤1+∥∇ϕ∥∞p¯BR+∫B2Rdz|z|N+(s−1)p=1+∥∇ϕ∞p¯BR+NB1(2R)(1−s)p−(1−s)p−.

This yields

∫BRvn(x)t(x)∫BR|ϕ(x)−ϕ(y)|p(x,y)|x−y|N+sp(x,y)dydx≤C4(R)∫BRvn(x)t(x)dx.

Using this, (4.57) and (4.59), we obtain from (4.55) that

(4.60)In≤C5min{λnp¯,λnp−}1Rsp−+C6(R)∑t∈{p¯,p∗} ∫BR |vn(x)|t(x)x.

Combining this with (4.56) and the boundedness of {v_n} in Ws,p(⋅,⋅)(RN) , we deduce from (4.54) that

1≤C7(R)min{λnp¯,λnp−}

and hence

λn≤C8(R),∀n∈N.

Thus {λ_n is a bounded sequence in ℝ and hence, up to a subsequence, we may assume that there exists λ* ∈ [0, ∞) such that

(4.61)limn→∞λn=λ∗.

Suppose that λ* > 0. From (4.53) and (4.60) we obtain

1≤(1+ε)∫RNϕ(x)λnp⋅(x)+ϕ(x)λnpvn(x)p¯+∫RNvn(x)−vn(y)p(x,y)|x−y|N+sp(x,y)dydx+C5minλnp¯,λnp1Rsp+C6(R)∑t∈{p¯,p⋅}∫BRvn(x)t(x)dx.

Letting n → ∞ in the last inequality, noticing (4.49), (4.61) and limn→∞∫BR |vn(x)|t(x)dx=0 for t∈{p¯,p∗} (see (4.46)), we obtain

1≤(1+ε)∫RN ϕλ∗p∗(x)+ϕλ∗p¯dμ¯+C5min{λ∗p¯,λ∗p−}Rsp−.

Letting R → ∞ and then letting ε→0+ , we deduce from the last inequality that

1≤∫RN ϕλ∗p∗(x)+ϕλ∗p¯dμ¯.

Invoking Proposition 2.2, we easily obtain from the last estimate that

λ∗≤21p−max||ϕ||Lμ¯p∗(⋅)(RN),||ϕ||Lμ¯p¯(RN).

From (4.48), (4.51), (4.61) and the last inequality, we arrive at

(4.62)Sq||ϕ||Lν¯q(⋅)(RN)≤21p−max||ϕ||Lμ¯p∗(⋅)(RN),||ϕ||Lμ¯p¯(RN).

If λ* = then by (4.48) and (4.51), we get ||ϕ||Lν¯q(⋅)(RN)=0 ; hence, (4.62) also holds. That is, (4.62) holds for any ϕ∈Cc∞(ℝN) and hence, (4.8) follows by invoking Proposition 6.1 and the definition of ν¯ (see (4.48)).

The fact that {xi}i∈I⊂C can be obtained by an argument similar to that of [27, Theorem 3.3] and we omit the proof. Next, we obtain the relation (4.9). Let i є l and for ρ>0 , define ψρ as in Lemma 4.4 with x₀ replaced by x_i. Thus ψρ∈C∞(ℝN) , 0≤ψρ≤1 , ψρ≡1 on Bρ(xi) , supp(ψρ)⊂B2ρ(xi) . Using (4.3) again, we have

Sq||ψρun||Lq(⋅)(ℝN)≤∥ψρun∥s,p.

Taking the limit inferior as n → ∞ in the above inequality and using (4.6) we obtain

(4.63)Sq∣ψρLvq(.)B2ρxi≤lim infn→∞ψρun∥s,p.

Hence,

(4.64)Sqlimsupρ→0+ ||ψρ||Lνq(⋅)(B2ρ(xi))≤limsupρ→0+ liminfn→∞ ∥ψρun∥s,p.

Invoking Proposition 2.2, we have

(4.65)∥ψρ∥Lνq(⋅)(B2ρ(xi))≥min(∫B2ρ(xi)|ψρ|q(x)dν)1qi,ρ+,(∫B2ρ(xi)|ψρ|q(x)dν)1qi,ρ−≥minν(Bρ(xi))1qi,ρ+,ν(Bρ(xi))1qi,ρ−,

where qi,ρ+:=maxx∈Bρ(xi)¯ q(x) , qi,ρ−:=minx∈Bρ(xi)¯ q(x) . Thus, we obtain a lower bound of the left-hand side of (4.64) as follows:

(4.66)lim supρ→0+||ψρ||Lνq(⋅)(B2ρ(xi))≥νi1q(xi)=νi1p¯s∗

due to the continuity of q and the fact that xi∈C . To obtain an upper bound of the right-hand side of (4.64), we first prove that there exist ρ0∈(0,1) and λ0∈(0,∞) such that

(4.67)0<Sq2vi1qx1≤lim infn→∞λn,ρ=:λ∗,ρ≤λ0 for any ρ∈0,ρ0,

where λn,ρ:=∥ψρun∥s,p. Indeed, by the continuity of q and the positiveness of ν_i, we can choose ρ0∈(0,1) such that

(4.68)Sqminνi1qi,ρ+,νi1qi,ρ−>Sq2νi1q(xi),∀ρ∈(0,ρ0)

From (4.63), (4.65) and (4.68), we infer Sq2νi1q(xi)≤λ*,ρ for all ρ∈(0,ρ0). On the other hand, by choosing ρ0 smaller if necessary we have

(4.69)limsupn→∞∫ℝN∫ℝN|un(y)|p(x,y)|ψρ(x)−ψρ(y)|p(x,y)|x−y|N+sp(x,y) yx<1, ∀ρ∈(0,ρ0)

in view of Lemma 4.4. Note that

(4.70)∫RNψρunp¯ dx+∫RN∫RNψρun(x)−ψρun(y)p(x,y)|x−y|N+sp(x,y)dy dx≤∫RNψρunp¯ dx+2p¯−1∫RN∫RNψρ(x)p(x,y)un(x)−un(y)p(x,y)|x−y|N+sp(x,y)dy dx+2p¯−1∫RN∫RNun(y)p(x,y)ψρ(x)−ψρ(y)p(x,y)|x−y|N+sp(x,y)dy dx.

Using (4.69), (4.70), the boundedness of {u_n} in Ws,p(⋅,⋅)(RN) and invoking Proposition 3.1, we can easily show that there exists λ0∈(0,∞) such that λn,ρ<λ0 for all n ϵ ℕ and ρ∈(0,ρ0) . Thus, (4.67) has been proved. Next, let ε>0 be arbitrary and fixed. We have

1=∫RNψρunλn,ρp¯ dx+∫RN∫RNψρun(x)−ψρun(y)p(x,y)λn,ρp(x,y)|x−y|N+sp(x,y)dy dx=∫RNψρunλn,ρp¯ dx+2∫B2pxi∫RN∖B2ρxiψρun(x)−ψρun(y)p(x,y)λn,ρp(x,y)|x−y|N+sp(x,y)dy dx+∫B2ρxi∫B2pxiψρun(x)−ψρun(y)p(x,y)λn,ρp(x,y)|x−y|N+sp(x,y)dy dx.

Hence, by utilizing (4.52) again we have

1≤∫RNψρunλn,ρp¯ dx+2∫B2pxi∫RN∖B2pxjun(x)λn,ρp(x,y)ψρ(x)−ψρ(y)p(x,y)|x−y|N+sp(x,y)dy dx+(1+ε)∫B2ρxi∫B2px1ψρ(x)p(x,y)un(x)−un(y)p(x,y)λn,ρp(x,y)|x−y|N+sp(x,y)dy dx+C(ε)∫B2px1∫B2pxiun(y)λn,ρp(x,y)ψρ(x)−ψρ(y)p(x,y)|x−y|N+sp(x,y)dy dx.

Combining this with the fact that 0≤ψρ≤1 yields

(4.71)1≤C(ε)minλn,ρp,λn,ρpi,ρ∫RN∫RNun(x)p(x,y)ψρ(x)−ψρ(y)p(x,y)|x−y|N+sp(x,y)dy dx+1+εminλn,ρp¯,λn,ρpi∫RNψρ(x)Un(x)dx,

where pi−=inf(x,y)∈B2ρ(xi)×B2ρ(xi)p(x,y) . Here and in what follows, for brevity we denote

(4.72)Un(x):=|un(x)|p¯+∫RN|un(x)−un(y)|p(x,y)|x−y|N+sp(x,y)dy,∀x∈RN,∀n∈N.

Using (4.67), we deduce from (4.71) that

1≤C(ϵ)min{λ∗,ρp¯,λ∗,ρpi−}lim supn→∞∫RN∫RNun(x)p(x,y)|ψρ(x)−ψρ(y)|p(x,y)|x−y|N+sp(x,y)dydx+1+ϵmin{λ∗,ρp¯,λ∗,ρpi−}∫RNψρdμ,∀ρ∈(0,ρ0).

Hence,

minλ⋆,ρp¯,λ⋆,ρpt−≤C(ε)lim supn→∞∫R2Nun(x)p(x,y)ψρ(x)−ψρ(y)p(x,y)|x−y|N+5p(x,y)dy dx+(1+ε)∫RNψρdμ.

Taking limit superior as ρ→0+ in the last inequality and invoking Lemma 4.4, we obtain

λ∗p¯≤(1+ε)μii.e.,λ∗≤(1+ε)1p¯μi1p¯,

where λ*:=limsupρ→0+ λ*,ρ and μi:=limρ→0+ μ(B2ρ(xi)) . Combining this with (4.64) and (4.66) together with the fact that ε was chosen arbitrarily we obtain (4.9). Hence, {xi}i∈I are also atoms of 𝜇.

Finally, to obtain (4.7) we note that for each ϕ∈C0(ℝN) , ϕ≥0 , the functional

u↦∫RNϕ(x)|u(x)|p¯+∫RN|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dydx

is convex and differentiable on Ws,p(⋅,⋅)(RN) . From this and (4.5) we infer

∫RNϕ(x)|u(x)|p¯+∫RN|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dydx≤lim infn→∞∫RNϕ(x)un(x)p¯+∫RNun(x)−un(y)p(x,y)|x−y|N+sp(x,y)dydx=∫RNϕdμ.

Thus,

μ≥|u|p+∫RN|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dy.

Extracting 𝜇 to its atoms, we get (4.7) and the proof is complete.

We conclude this section by proving Theorem 4.2.

Proof of Theorem 4.2. For each R > 0, define ϕ_R as in Lemma 4.5. Thus ϕR∈Cc∞(ℝN) , 0≤ϕR≤1 , ϕR≡0 on B_R and ϕR≡1 on B2Rc , and ||∇ϕR||∞≤2R. In order to obtain (4.13), we decompose

(4.73)∫RNUn(x)dx=∫RNϕR(x)Un(x)dx+∫RN1−ϕR(x)Un(x)dx,

where U_n is given by (4.72). By (4.11) and the fact that

∫B2RcUn(x)dx≤∫RNϕR(x)Un(x)dx≤∫BRcUn(x)dx

for all n ϵ N and R > 0, we obtain

(4.74)μ∞=limR→∞lim supn→∞∫RNϕR(x)Un(x)dx.

On the other hand, the fact that 1−ϕR∈Cc∞(RN) gives

(4.75)limn→∞∫RN1−ϕR(x)Un(x)dx=∫RN1−ϕR(x)dμ.

Meanwhile,

limR→∞∫RNϕR(x)dμ=0

in view of the Lebesgue dominated convergence theorem. From the last two equalities, we obtain

limR→∞limn→∞∫RN1−ϕR(x)Un(x)dx=μRN.

From this and (4.73)-(4.75) we obtain (4.13).

In order to prove (4.12), we decompose

(4.76)∫RNun(x)q(x)dx=∫RNϕRq(x)un(x)q(x)dx+∫RN1−ϕRq(x)un(x)q(x)dx.

From the definition (4.10) of ν_∞ and the estimate

∫B2Rcun(x)q(x)dx≤∫RNϕRq(x)un(x)q(x)dx≤∫BRcun(x)q(x)dx

for all n ϵ N and R > 0, we deduce

(4.77)v∞=limR→∞lim supn→∞∫RNϕRq(x)(x)un(x)q(x)dx.

Arguing as that obtained (4.13) above for which ϕ_R is replaced with ϕRq(x) , we obtain (4.12).

We conclude the proof by proving (4.14). Without loss of generality we assume ν∞>0. Let ε∈(0,1) be arbitrary and fixed. By (ε_∞), we can choose R₁ > 1 such that

(4.78)p(x,y)−p¯|<εand|q(x)−q∞|<εfor all|x|,|y|>R1.

From (4.3), we have

(4.79)Sq||ϕRun||Lq(⋅)(RN)≤||ϕRun||s,p.

For R > R₁, using (4.78) and Proposition 2.2 we have

∥ϕRun∥Lq(⋅)(RN)=∥ϕRun∥Lq(⋅)(BRc)≥min(∫BRcϕRq(x)|un|q(x)dx)1q∞+ϵ,(∫BRcϕRq(x)|un|q(x)dx)1q∞−ϵ≥min(∫B2Rc|un|q(x)dx)1q∞+ϵ,(∫B2Rc|un|q(x)dx)1q∞−ϵ.

Thus,

(4.80)lim infR→∞lim supn→∞∥ϕRun∥Lq(⋅)(RN)≥minν∞1q∞+ϵ,ν∞1q∞−ϵ.

Next, we estimate the right-hand side of (4.79). To this end, denote σn,R:=∥ϕRun∥s,p for brevity. We will show that there exist R2∈(R1,∞) and σ∈(0,∞) such that

(4.81)0<Sq14v∞14ω∞≤0∗,R:=limn→∞supn→R<0,∀R∈R2,∞.

Indeed, we first choose ε¯>0 sufficiently small such that

(4.82)min{(ν∞2)1q∞+ε¯,(ν∞2)1q∞−ε¯}>(ν∞4)1q∞.

Then we can find R¯2>R1 such that

(4.83)∥ϕRun∥Lq(⋅)(RN)≥min{(∫BRcϕRq(x)|un|q(x)dx)1q∞+ϵ¯,(∫BRcϕRq(x)|un|q(x)dx)1q∞−ϵ¯}

for all R>R¯2 . Finally, by (4.77), we can find R2>R¯2 such that

(4.84)lim supn→∞∫RNϕRq(x)unq(x)dx=lim supn→∞∫BRcϕRq(x)unq(x)dx>v∞2

for all R>R¯2 . From (4.83) and (4.84) we get

lim supn→∞ϕRunLq(⋅)RN≥minv∞214∞+z¯,V∞21q∞−ε¯

and hence, by (4.82),

lim supn→∞ϕRunLq(⋅)RN≥V∞41q∞

for all R>R2 . This and (4.79) yield Sq(14ν∞)1q∞≤σ∗,R for all R∈(R2,∞) . By a similar argument to that obtained (4.67), invoking Lemma 4.5 and choosing R₂ larger if necessary, we can show that there exists σ∈(0,∞) such that σ∗,R<σ for all R∈(R2,∞) . Thus, (4.81) has been proved.

We now turn to estimate the right-hand side of (4.79). For each R>R2 given, let nk=nk(R) (k=1,2,⋯) be a sequence such that

(4.85)limk→∞σnk,R=lim supn→∞σn,R=σ⋆,R.

Utilizing Proposition 3.1 and (4.52) again, we have

1=∫RNϕR(x)unk(x)p¯σnk,Rp¯ dx+∫RN∫RNϕRunk(x)−ϕRunk(y)p(x,y)σnk,Rp(x,y)|x−y|N+sp(x,y)dx dy=∫RNϕR(x)p¯unk(x)p¯σnk,Rp¯ dx+2∫BRc∫BRϕR(x)p(x,y)unk(x)p(x,y)σnk,Rp(x,y)|x−y|N+sp(x,y)dy dx+∫BRc∫BRcϕRunk(x)−ϕRunk(y)p(x,y)σnkp(x,y)|x−y|N+sp(x,y)dy dx≤∫BRcϕR(x)p¯unk(x)p¯σnk,Rp¯ dx+2∫BRc∫BRunk(x)p(x,y)σnk,Rp(x,y)ϕR(x)−ϕR(y)p(x,y)|x−y|N+sp(x,y)dy dx+C(ε)∫BRc∫BRcunk(x)p(x,y)σnk,Rp(x,y)ϕR(x)−ϕR(y)p(x,y)|x−y|N+sp(x,y)dy dx+(1+ε)∫BRc∫BRcϕR(y)p(x,y)σnk,Rp(x,y)unk(x)−unk(y)p(x,y)|x−y|N+sp(x,y)dx dy.

This and the fact that 0≤ϕR≤1 yield

1≤C(ε)minσnk,Rp¯,σnk,Rp−∫BRc∫RNunk(x)p(x,y)ϕR(x)−ϕR(y)p(x,y)|x−y|N+sp(x,y)dy dx+1+εminσnk,Rp¯+ε,σnk,Rp¯−ε∫BRcϕR(x)Unk(x)dx.

Taking limit superior as k → ∞ in the last inequality with noticing (4.81) and (4.85) we obtain

(4.86)≤C(ε)minσ⋆,Rp¯,σ⋆,Rp−lim supn→∞∫BRc∫RNun(x)p(x,y)ϕR(x)−ϕR(y)p(x,y)|x−y|N+sp(x,y)dy dx+1+εminσ⋆,Rp¯+ε,σ⋆,Rp¯−εlim supn→∞∫BRcϕR(x)Un(x)dx.

Now, taking the limit as R → ∞ in (4.86) with taking Lemma 4.5 and (4.74) into account, we deduce

1≤1+εminσ∗p¯+ε,σ∗p¯−εμ∞, i.e., σ⋆≤(1+ε)1p−εmaxμ∞1p¯+εμ∞1p¯+ε,

where σ∗:=lim infR→∞σ∗,R and hence, 0 < σ* < σ due to (4.81). From this, (4.79) and (4.80) we obtain

Sqminv∞1q∞+ε,v∞1q∞−ε≤(1+ε)1p¯−εmaxμ∞1p+ε¯,μ∞1−μ¯−ε.

Since ε was chosen arbitrarily in the last inequality, (4.14) follows. The proof of Theorem 4.2 is complete.□

5 Application

5.1 The existence of solutions

In this section, we investigate the existence and multiplicity of solutions to the following problem

(5.1)Lu+|u|p(x)−2u=f(x,u)+λ|u|q(x)−2u in RN,u∈Ws,p(⋅,⋅)RN,

where s,p,q satisfy (𝒫₂), (𝒬₂) and (ε_∞) with p⁺ < q⁻; the operator 𝓛 is defined as in (1.2); 𝜆 is a positive real parameter; and the nonlinear term f satisfies the following assumptions.

(𝓕1) f:RN×R→R is a Carathéodory function such that f is odd with respect to the second variable.

(𝓕2) There exist functions r_j,a_j with rj∈C+(RN) , infx∈RN[q(x)−rj(x)]>0, aj∈L+q(⋅)q(⋅)−rj(⋅)(RN)(j=1,⋯,m) , and max1≤j≤mrj+>p− such that

|f(x,u)|≤∑j=1m aj(x)|u|rj(x)−1fora.e.x∈RNand allu∈R.

(𝓕3) There exist Bε(x0) and a∈L+q(⋅)q(⋅)−p+(Bε(x0)) such that sup|u|≤M|F(x,u)|∈L1Bεx0 for each M > 0, and

lim|u|→∞F(x,u)a(x)|u|p+=∞uniformly for a.e.x∈Bε(x0),

where F(x,u):=∫0uf(x,τ)dτ.

(𝓕4) There exist α∈[p+,q−) and g∈L+1(ℝN) such that

αF(x,u)−f(x,u)u≤g(x) for a.e. x∈ℝN and all u∈ℝ.

A trivial example for f(x,u) satisfying (F1)−(F4) is f(x,u)=a(x)|u|r(x)−2u with r∈C+(ℝN) such that p+<r− and infx∈ℝN[q(x)−r(x)]>0, and a∈Lq(⋅)q(⋅)−r(⋅)(RN) with a(x)>0 a.e. on some ball B∈ℝN.

We say that u∈Ws,p(⋅,⋅)(RN) is a (weak) solution of problem (5.1) if

By Theorems 3.3 and 3.4, this definition is clearly well defined under assumptions (F1)−(F2). Our main existence result is stated as follows.

Theorem 5.1

Let (𝒫₂), (𝒬₂) and (ε_∞) hold with p+<q−. If (F1)−(F4) are fulfilled, then there exists a sequence {λk}k=1∞ of positive real numbers with λk+1<λk for all k ϵ ℕ such that for any λ∈(λk+1,λk), problem (5.1) admits at least k pairs of nontrivial solutions.

5.2 Proof of Theorem 5.1

In order to prove Theorem 5.1, we will make use of the following abstract result for symmetric C¹ functionals, which is a variant of Theorem 2.19 in [33] (see also [34, Theorem 10.20]).

Lemma 5.2

([33]) Let E=V⊕X, where E is a real Banach space and V is finite dimensional. Suppose that J∈C1(E,ℝ) is an even functional satisfying J(0)=0 and

(𝓙1) there exist constants ρ, β>0 such that J(u)≥β for all u∈∂Bρ∩X;

(𝓙2) there exists a subspace Ẽ of E with dimV<dimE˜<∞ and {u∈E˜: J(u)≥0} is bounded in E;

(𝓙3) for 𝛽 and Ẽ respectively given in (𝓙1) and (𝓙2), J satisfies the (PS)_c condition for any c∈[0,L] with L:=supu∈E˜ J(u).

Then J possesses at least dimE˜−dimV pairs of nontrivial critical points.

Proof

The proof is similar to that of [34, Theorem 10.20] for which we take Em in the proof of [34, Lemma 10.19] as Em=span{e1,⋯,em}, where {ek}k=1dimE~ is a basis of Ẽ.

To determine solutions to problem (5.1), we will apply Lemma 5.2 for E:=Ws,p(⋅,⋅)(RN) endowed with the norm ∥⋅∥:=∥⋅∥s,p and J=Jλ, where Jλ: E→ℝ is the energy functional associated with problem (5.1) defined as

It is clear that under the assumptions (F1)−(F2), J_λ is of class C1(E,ℝ) and its Fréchet derivative Jλ':E→E∗ is given by

Here, E∗ and 〈⋅,⋅〉 denote the dual space of E and the duality pairing between E and E∗, respectively. Clearly, J_λ is even in E, Jλ(0)=0, and each critical point of J_λ is a solution to problem (5.1). The next lemma will be utilized for verifying (𝓙3).

Lemma 5.3

For any given λ>0, J_λ satisfies the (PS)_c condition provided

(5.4)c<1α−1q−minSq(qh)+,Sq(qh)−minλ−h+,λ−h−−∥g∥1α,

where h(x):=p¯q(x)−p¯ for x ∈ ℝ^N and Sq is defined as in (4.3).

Proof

Let {u_n} be a (PS)_c sequence for J_λ with c satisfying (5.4). We first claim that {u_n} is bounded in E. Indeed, by (F4) and invoking Proposition 3.1 we have that for n large,

c+1+un≥Jλun−1αJλ′un,un≥∫RN∫RN1p(x,y)−1αun(x)−un(y)p(x,y)|x−y|N+sp(x,y)dx dy+∫RN1p¯−1α|u|p¯ dx+λ∫RN1α−1q(x)|un|q(x)x+∫RN1αf(x,un)un−F(x,un)x≥λ1α−1q−∫RN|un|q(x) dx−1α∥g∥1.

That is,

(5.5)λ1α−1q−∫RNunq(x)dx≤C1+un, for all n∈N large. .

Here and in the remaining proof, C_i (i є ℕ) denotes a positive constant independent of n. On the other hand, by the relation between modular and norm (see Proposition 3.1) and (𝓕2) we have that for n large,

1p¯unp−−1≤Jλun+∫RNFx,undx+λ∫RN1q(x)unq(x)dx≤c+1+∑j=1m∫RNaj(x)rj(x)unrj(x)dx+λq−∫RNunq(x)dx.

Then, by the Young inequality we easily get

(5.6)∥un∥p−≤C21+∫RN|un|q(x)x.

From (5.5) and (5.6) we obtain

unp−≤C31+un, for all n∈N large.

This implies the boundedness of {u_n} since p−>1 and hence,

(5.7)C∗:=supn∈N∫RN∫RN|un(x)−un(y)|p(x,y)|x−y|N+sp(x,y)dx dy+∫RN|un|p¯ dx<∞

in view of Proposition 3.1. Then, invoking Theorems 3.2, 4.1 and 4.2, up to a subsequence, we have

(5.8)un(x)→u(x) for a.e. x∈ℝN,

(5.9)un⇀u in E,

(5.10)Un(x)⇀∗μ≥U(x)+∑i∈IμiδxiinM(RN),

(5.11)|un|q(x)⇀∗ν=|u|q(x)+∑i∈Iνiδxi in M(ℝN),

(5.12)Sq νi1p¯s∗≤μi1p¯, ∀i∈I,

where Un(x):=un(x)p¯+∫RNun(x)−un(y)p(x,y)|x−y|N+sp(x,y)dy and U(x):=|u(x)|p¯+∫RN|u(x)−u(y)|p(x,y)|x−y|N+sp(x,y)dy for n ϵ ℕ and x ∈ ℝ^N. Moreover, we have

(5.13)limsupn→∞∫ℝNUn(x)x=μ(ℝN)+μ∞,

(5.14)lim supn→∞∫RN|un|q(x)dx=v(RN)+v∞,

(5.15)Sqν∞1q∞≤μ∞1p¯.

We will show that I=∅ and ν∞=0. For this purpose we invoke (F4) to estimate

c=limn→∞[Jλ(un)−1α〈Jλ′(un),un〉]≥(1α−1q−)λ limsupn→∞∫ℝN|un|q(x)x−∥g∥1α.

Combining this with (5.14) gives

(5.16)c≥(1α−1q−)λ[ν(ℝN)+ν∞]−∥g∥1α.

We now suppose on the contrary that I=∅. Let i є l and for ρ>0, define ψρ as in Lemma 4.4 with x0 replaced by xi. For an arbitrary and fixed 𝜌, it is not difficult to see that {unψρ} is a bounded sequence in E. Hence,

on(1)=⟨J′λ(un),unψρ⟩=∫RNψρUnx−λ∫RNψρ|un|q(x)dx−∫RNf(x,un)unψρx+∫RN∫RNun(x)−un(y)p(x,y)−2un(x)−un(y)un(y)ψρ(x)−ψρ(y)|x−y|N+sp(x,y)dx dy.

This yields

where

I1(n,ψρ):=∫ℝNf(x,un)unψρx

and

I2(n,ψρ):=∫RN∫RN|un(x)−un(y)|p(x,y)−2(un(x)−un(y))un(y)(ψρ(x)−ψρ(y))|x−y|N+sp(x,y)dx dy.

Note that the boundedness of {u_n} in E implies the boundedness of {u_n} in Lq(⋅)(RN) due to Theorem 3.3. Hence, from (𝓕2) and invoking Propositions 2.2 and 2.3 we have

(5.18)I1n,ψρ≤∑j=1m∫RNaj(x)unrj(x)ψρdx≤∑j=1m2∥aj∥Lq(⋅)q(⋅)−rj(⋅)(B2ρ(xi))∥|un|rj∥Lq(⋅)rj(⋅)(B2ρ(xi))≤∑j=1m2∥aj∥Lq(⋅)q(⋅)−rj(⋅)(B2ρ(xi))1+∥un∥Lq(⋅)(RN)rj+≤C4∑j=1m∥aj∥Lq(⋅)q(⋅)−rj(⋅)(B2ρ(xi)),∀n∈N.

Here and in the remaining proof, Ci(i∈N) denotes a positive constant independent of n and ρ From (5.18), we obtain

(5.19)limsupρ→0+lim supn→∞I1n,ψρ=0.

In order to estimate I2(n,ψρ), let δ > 0 be arbitrary and fixed. By (5.7) and the Young inequality we have

(5.20)I2n,ψρ≤δ∫RN∫RNun(x)−un(y)p(x,y)|x−y|N+sp(x,y)dx dy+C5∫RN∫RNun(y)p(x,y)ψρ(x)−ψρ(y)p(x,y)|x−y|N+sp(x,y)dx dy≤C∗δ+C5∫RN∫RNun(y)p(x,y)ψρ(x)−ψρ(y)p(x,y)|x−y|N+sp(x,y)dx dy.

Taking limit superior in (5.20) as n → ∞ then taking limit superior as ρ → 0⁺ with taking Lemma 4.4 into account, we arrive at

limsupρ→0+lim supn→∞I2n,ψρ≤C⋆δ.

Since δ > 0 was chosen arbitrarily we obtain

(5.21)limsupρ→0+lim supn→∞I2n,ψρ=0.

Now, by taking limit superior in (5.17) as ρ → 0⁺ with taking (5.19) and (5.21) into account, we obtain

μi=λνi.

Plugging this into (5.12) we get

μi≥Sq(xi)p¯q(xi)−p¯λ−p¯q(xi)−p¯.

This yields

(5.22)λνi=μi≥min{S(qh)+,S(qh)−}min{λ−h+,λ−h−}.

From (5.22) and (5.16), we obtain

c≥1α−1q−λvi−∥g∥1α≥1α−1q−minS(qh)+,S(qh)−minλ−h+,λ−h−−∥g∥1α,

which is a contradiction to (5.4), and hence, I = ∅. Next, we prove that ν∞=0. Suppose on the contrary that ν∞>0. Let ϕ_R be the same as in Lemma 4.5. Using a similar argument to that obtained (5.17), we arrive at

(5.23)lim supn→∞∫RNϕRUn dx≤λlim supn→∞∫RNϕRunq(x)dx+lim supn→∞I3n,ϕR+lim supn→∞I4n,ϕR,

and

(5.24)λlim supn→∞∫RNϕRunq(x)dx≤lim supn→∞∫RNϕRUn dx+lim supn→∞I3n,ϕR+lim supn→∞I4n,ϕR,

where

I3n,ϕR:=∫RNfx,ununϕR dx

and

I4n,ϕR:=∫RN∫RNun(x)−un(y)p(x,y)−2un(x)−un(y)un(y)ϕR(x)−ϕR(y)|x−y|N+sp⁡(x,y)dx dy.

We claim that

(5.25)limR→∞lim supn→∞|I3(n,ϕR)|=0

and

(5.26)limR→∞lim supn→∞|I4(n,ϕR)|=0.

Indeed, the equality (5.25) can be obtained in a similar fashion to (5.19). To prove (5.26), we proceed as in (5.20) to get

I4n,ϕR≤C∗δ+C5∫RN∫RNun(y)p(x,y)ϕR(x)−ϕR(y)p(x,y)|x−y|N+sp(x,y)dx dy

for each δ > 0 arbitrary and fixed. Taking limit superior in the last estimate as n → ∞ and then taking limit as R → ∞ with taking Lemma 4.5 into account, we obtain

limR→∞lim supn→∞|I4(n,ϕR)|≤C∗δ.

Since δ > 0 in the last inequality can be taken arbitrarily we deduce (5.26). Using (5.25) and (5.26) and letting R → ∞ in (5.23) and (5.24) we obtain

(5.27)μ∞=λν∞.

Here we have used (4.74) and the fact that

v∞=limR→∞lim supn→∞∫RNunq(x)ϕR dx,

which can be seen by using ϕ_R in place of ϕRq(x) in (4.76). Combining (5.27) with (5.15) gives

(5.28)λν∞=μ∞≥Sq∞p¯q∞−p¯λ−p¯q∞−p¯.

The fact that q∞=lim|x|→∞⁡q(x)∈[q−,q+] yields

(qh)−≤q∞p¯q∞−p¯≤(qh)+and h−≤p¯q∞−p¯≤h+.

From this and (5.28) one has

λν∞≥min{S(qh)+,S(qh)−}min{λ−h+,λ−h−}.

Utilizing this estimate we deduce from (5.16) that

c≥1α−1q−λν∞−∥g∥1α≥1α−1q−min{S(qh)+,S(qh)−}min{λ−h+,λ−h−}−∥g∥1α,

which is a contradiction to (5.4), and hence; ν∞=0.

Combining the facts that I = ∅ and ν∞=0. with (5.11) and (5.14), we obtain

lim supn→∞∫RNunq(x)dx=∫RN|u|q(x)dx.

Invoking the Fatou lemma we get from (5.8) that

∫RN|u|q(x)dx≤lim infn→∞∫RNunq(x)dx.

Thus,

limn→∞∫RNunq(x)dx=∫RN|u|q(x)dx.

By a Brezis-Lieb type result for the Lebesgue spaces with variable exponents (see e.g., [27, Lemma 3.9]), it follows from the last equality and (5.8) that

∫ R N u n − u q ( x ) d x → 0 , i.e., u n → u in L q ( ⋅ ) R N .

Consequently, we have ∫ R N | u n | q ( x ) − 2 u n ( u n − u ) d x → 0 by invoking Proposition 2.3 and the boundedness of {u_n} in L q ( ⋅ ) ( R N ) . Also, we easily obtain ∫ R N f x , u n u n − u d x → 0 by using (𝓕2), (5.9), Proposition 2.3 and Theorem 3.4. We therefore have

Hence, u_n→u in E in view of [9, Lemma 4.2 (i)]. The proof is complete. □

We now conclude this section by proving Theorem 5.1.

Proof of Theorem 5.1. We will show that conditions ( J 1 ) − ( J 3 ) of Lemma 5.2 are fulfilled with E := W s , p ( ⋅ , ⋅ ) ( R N ) and J = J_𝜆. In order to verify (ℐ1), let { e n } n = 1 ∞ be a Schauder basis of E and let { e n ∗ } n = 1 ∞ ⊂ E ∗ be such that for each n ϵ ℕ,

e n ∗ ( u ) = α n for u = ∑ k = 1 ∞ α k e k ∈ E .

For each k ϵ ℕ, define

V k := u ∈ E : e n ∗ ( u ) = 0 , ∀ n > k , X k := u ∈ E : e n ∗ ( u ) = 0 , ∀ n ≤ k ,

and

(5.29) ξ k := sup u ∈ X k ∥ u ∥ ≤ 1 max 1 ≤ j ≤ m ∥ u ∥ L r j ( ⋅ ) ( a j , R N ) .

Then,

E = V k ⊕ X k , ∀ k ∈ N .

Since X k + 1 ⊂ X k (k ϵ ℕ), we have

0 ≤ ξ k + 1 ≤ ξ k , ∀ k ∈ N .

Thus, the sequence {ξ_k} converges to some ξ_* as k → ∞. We claim that ξ_* = 0. Indeed, for each k ϵ ℕ there exists u k ∈ X k such that ∥ u k ∥≤ 1 and

(5.30) 0 ≤ ξ k ≤ max 1 ≤ j ≤ m ∥ u k ∥ L r j ( ⋅ ) ( a j , R N ) + 1 k .

Since {u_k} is bounded in E, up to a subsequence we have

(5.31) u k ⇀ u in E

and hence, by Theorem 3.4,

(5.32) u k → u in L r j ( ⋅ ) a j , R N , ∀ j ∈ { 1 , ⋯ , m } .

From (5.31) and the definition of X_k we have that for any n ϵ ℕ,

(5.33) 〈 e n ∗ , u 〉 = lim k → ∞ 〈 e n ∗ , u k 〉 = 0.

This yields u = 0. This fact together with (5.32) and (5.30) infer ξ_* = 0. That is, we have just proved that

(5.34) lim k → ∞ ξ k = 0.

For u ∈ X_k with ∥ u ∥= ρ k > 1 , by (𝓕2) and invoking Proposition 2.2 and Theorem 3.4 we have

(5.35) J λ ( u ) ≥ 1 p ¯ ∥ u ∥ p − − 1 − ∑ j = 1 m 1 r j − ∫ R N a j ( x ) | u | r j ( x ) d x − λ q − ∫ R N | u | q ( x ) d x ≥ 1 p ¯ ∥ u ∥ p − − 1 p ¯ − 1 r − ∑ j = 1 m ∥ u ∥ L r j ( ⋅ ) ( a j , R N ) r + + 1 − λ q − max ∥ u ∥ L q ( ⋅ ) ( R N ) q + , ∥ u ∥ L q ( ⋅ ) ( R N q − ≥ 1 p ¯ ∥ u ∥ p − − m r − ξ k r + ∥ u ∥ r + − 1 p ¯ + m r − − max S q − q + , S q − q − q − λ ∥ u ∥ q + ,

where r − := min 1 ≤ j ≤ m r j − and r + := max 1 ≤ j ≤ m r j + . Let ρ k > 0 be such that

m r − ξ k r + ρ k r + = 1 2 p ¯ ρ k p − i .e ., ρ k = r − 2 m p ¯ ξ k r + 1 r + − p − .

Note that ξ_k → 0 as k → ∞ by (5.34) and hence, ρk→∞ as k → ∞. Thus, we can fix k0∈ℕ such that

(5.36) ρ k 0 > 1 and 1 2 p ¯ ρ k 0 p − − 1 p ¯ + m r − > 1 4 p ¯ ρ k 0 p − .

Then, (5.35) yields

(5.37) J λ ( u ) ≥ 1 4 p ¯ ρ k 0 p − − max S q − q + , S q − q − q − ρ k 0 q + λ , ∀ u ∈ X k 0 ∩ B ρ k 0 .

Therefore, by choosing V := V_k0, X :=X_k0 and λ ⋆ := q − ρ k 0 p − − q + 4 p ¯ max S q − q + , S q − q − , we have that for any λ ∈ (0,λ_*),

J λ ( u ) ≥ β , ∀ u ∈ X ∩ B ρ

with ρ = ρ_k₀ and β = max S q − q + , S q − q − ρ k 0 q + q − λ ⋆ − λ . That is, J_λ verifies (ℐ1) in Lemma 5.2.

Next, we show that J_λ verifies (ℐ2) and (ℐ3) in Lemma 5.2. Let ( y k , φ k ) be the k^th eigenpair of the following eigenvalue problem

− Δ u = y u in B ε x 0 , u = 0 on ∂ B ε x 0 .

Extend φ k ( x ) to ℝ^N by putting φ k ( x ) = 0 for x ∈ R N ∖ B ε ( x 0 ) . Clearly, { φ k } ⊂ E . Define

E k := span φ 1 , ⋯ , φ k ( k ∈ N ) .

Let k ϵ ℕ be arbitrary and fixed. We claim that there exists R_k > 0 independent of 𝜆 such that

(5.38) J λ ( u ) ≤ 0 , ∀ u ∈ E k ∖ B R k .

Indeed, since all norms on E_k are equivalent we can find ζ k > 0 such that

(5.39) ζ k ∥ u ∥ ≤ ∥ u ∥ L p ¯ ( a , B ϵ ( x 0 ) ) , ∀ u ∈ E k .

Choose θ_k > 0 such that

(5.40) 1 p − − θ k ζ k p ¯ < 0.

By (𝓕3), we can choose M k > 0 such that

F ( x , u ) ≥ θ k a ( x ) | u | p ¯ for a .e . x ∈ B ε ( x 0 ) and all | u | ≥ M k .

This infers

(5.41) F ( x , u ) ≥ θ k a ( x ) | u | p ¯ − sup | u | ≤ M k | F ( x , u ) | for a.e. x ∈ B ε x 0 and all u ∈ R .

From (5.40), (5.41) and invoking Proposition 2.2 again, we have that for u∈Ek with ∥ u ∥ ≥ R k > 1 ,

J λ ( u ) ≤ 1 p − ∥ u ∥ p ¯ − θ k ∫ B ϵ ( x 0 ) a ( x ) | u | p ¯ d x + ∫ B ϵ ( x 0 ) sup | t | ≤ M k | F ( x , t ) | d x ≤ 1 p − − θ k ζ k p ¯ ∥ u ∥ p ¯ + ∫ B ϵ ( x 0 ) sup | t | ≤ M k | F ( x , t ) | d x < 0

provided R_k large enough. Clearly, R_k can be chosen independently of 𝜆. That is, we have just obtained (5.38). Noting J λ ( 0 ) = 0 , we deduce from (5.38) that

It is clear that for all k ϵ ℕ, L_k is independent of 𝜆 and L_k ϵ [0, ∞) due to (𝓕2). Finally, let { λ k } k = 1 ∞ ⊂ ( 0 , λ ∗ ) be such that for any k ϵ ℕ,

(5.42) L k 0 + k < 1 a − 1 q min S q ( q h ) + , S q ( q h ) − min λ k − h + , λ k − h − − ∥ g ∥ 1 α , λ k + 1 < λ k .

Then, for any λ∈(λk+1,λk) we have

Lk0+k<(1α−1q−)min{Sq(qh)+,Sq(qh)−}min{λ−h+,λ−h−}−∥g∥1α

and hence, J_λ satisfies the (PS)_c for any c ϵ [0, L_k₀+k] in view of Lemma 5.3. Thus, J_λ satisfies (ℐ2) and (ℐ3) with E˜=Ek0+k and L=Lk0+k. So, J_λ admits at least dim E˜−dim V=k pairs of nontrivial critical points in view of Lemma 5.2; hence, problem (5.1) has at least k pairs of nontrivial solutions. The proof is complete.

A An auxiliary result

In this appendix, we state a result for the Radon measures on ℝ^N, which is necessary for proving Theorem 4.1.

Proposition A.1

Let p, q, r ϵ C₊(ℝ^N) such that inf x ∈ R N { r ( x ) − max { p ( x ) , q ( x ) } } > 0. Let μ ν be two finite and nonnegative Radon measures on ℝ^N such that

∥ ϕ ∥ L v r ( · ) R N ≤ C max ∥ ϕ ∥ L μ p ( · ) R N , ∥ ϕ ∥ L μ q ( · ) R N , ∀ ϕ ∈ C c ∞ R N ,

for some constant C > 0. Then there exist an at most countable set { x i } i ∈ I of distinct point in ℝ^N and { ν i } i ∈ I (0, ∞) such that

ν=∑i∈Iνiδxi.

In order to prove Proposition 6.1, we will make use of the following result.

Lemma A.2

Let p,q,r∈C+(ℝN) such that infx∈ℝN{r(x)−max{p(x),q(x)}}>0. Let ν be a finite nonnegative Radon measure on ℝ^N such that

∥ ϕ ∥ L ν r ( ⋅ ) ( R N ) ≤ C max { ∥ ϕ ∥ L ν p ( ⋅ ) ( R N ) , ∥ ϕ ∥ L ν q ( ⋅ ) ( R N ) } , ∀ ϕ ∈ C c ∞ ( R N ) .

Then v = 0 or there exist { x i } i = 1 n of distinct points in ℝ^N and { ν i } i = 1 n ⊂ ( 0 , ∞ ) such that ν = ∑ i = 1 n ν i δ x i .

The proof of Lemma 6.2 is similar to that of [27, Lemma 3.8] and using this result we can prove Proposition 6.1 via the same method as in [21, Lemma 3.2] and we leave the proofs to the reader.

Acknowledgements

The first author was supported by University of Economics Ho Chi Minh City, Vietnam. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1F1A1057775).

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Received: 2020-01-19

Accepted: 2020-10-16

Published Online: 2020-12-08

This work is licensed under the Creative Commons Attribution 4.0 International License.

The concentration-compactness principles for W^s,p(·,·)(ℝ^N) and application

Abstract

1 Introduction

2 Variable exponent Lebesgue spaces and fractional Sobolev spaces

Proposition 2.1

Proposition 2.2

Proposition 2.3

Proposition 2.4

3 The Sobolev spaces W s , p ( ⋅ , ⋅ ) ( Ω )

Proposition 3.1

Theorem 3.2

Theorem 3.3

Proof

Theorem 3.4

4 The concentration-compactness principles for W s , p ( ⋅ , ⋅ ) ( R N )

4.1 Statements of the concentration-compactness principles

Theorem 4.1

Theorem 4.2

Example 4.3

4.2 Auxiliary lemmas and proofs of the concentration-compactness principles

Lemma 4.4

Lemma 4.5

5 Application

5.1 The existence of solutions

Theorem 5.1

5.2 Proof of Theorem 5.1

Lemma 5.2

Proof

Lemma 5.3

Proof

A An auxiliary result

Proposition A.1

Lemma A.2

Acknowledgements

References

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