Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

Sitong Chen; Xianhua Tang; Jiuyang Wei

doi:10.1515/anona-2020-0118

Open Access Published by De Gruyter May 30, 2020

Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

Sitong Chen , Xianhua Tang and Jiuyang Wei

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0118

Abstract

This paper deals with the following Choquard equation with a local nonlinear perturbation:

−Δu+u=Iα∗|u|α2+1|u|α2−1u+f(u),x∈R2;u∈H1(R2),

where α ∈ (0, 2), I_α : ℝ² → ℝ is the Riesz potential and f ∈ 𝓒(ℝ, ℝ) is of critical exponential growth in the sense of Trudinger-Moser. The exponent α2+1 is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponent α2+1 and the critical exponential growth of f(u).

Keywords: Choquard equation; Lower critical exponent; Nehari-type ground state solution; Critical exponential growth; Trudinger-Moser

MSC 2010: 35J20; 35J62; 35Q55

1 Introduction

In the past few years, the following Choquard equation:

−Δu+u=Iα∗|u|q|u|q−1u,x∈RN;u∈H1(RN), (1.1)

has attracted considerable attention, where N ≥ 1, α ∈ (0, N), 2 < q < 2^* and I_α : ℝ^N → ℝ is the Riesz potential. Physical motivation of (1.1) comes from the case that N = 3, α = 2 and q = 2. In this case, Eq.(1.1) is called the Choquard-Pekar equation [21, 31], Hartree equation [19] or Schrödinger-Newton equation [27, 39], depending on its physical backgrounds and derivations. The existence of a ground state in this case was studied in [21, 22, 28] via variational arguments. In a pioneering work, Lieb [21] first obtained the existence and uniqueness of positive solutions to (1.1) with N = 3, α = 2 and q = 2. Later, Lions [22, 23] got the existence and multiplicity results of normalized solution on the same topic. Moroz and Van Schaftingen [28] proved that (1.1) has a ground state solution if

N−2N+α<1q<NN+α;

and it has no nontrivial solution when either q≤αN+1 or q≥N+αN−2. The endpoints of the above interval are critical exponents. The upper critical exponent N+αN−2 plays a similar role as the Sobolev critical exponent in the local semilinear equations [9, 40]. The lower critical exponent αN+1 comes from the Hardy-Littlewood-Sobolev inequality. So far, there are a variety of interesting results concerning the existence of nontrivial solutions for more general Choquard equation with upper critical growth, see for example, see [6, 7, 17, 24] and the references cited therein. However, to the best of our knowledge, it seems that the only available works regarding the existence of nontrivial solutions for Choquard equation with lower critical exponent and a local nonlinear perturbation are the papers [35, 38, 41]. In details, Van Schaftingen and Xia [35] proved that the following Choquard equation:

−Δu+u=Iα∗|u|αN+1|u|αN−1u+f(u),x∈RN;u∈H1(RN) (1.2)

admits a ground state solution if there exists Λ > 0 such that f satisfies the following three assumptions:

f ∈ 𝓒(ℝ, ℝ), f(t) = o(|t|) as t → 0 and f(t) = o(|t|^2N/(N–2)) as |t| → ∞.
there exists μ > 2 such that 0 < μF(t) ≤ f(t)t for all t ≠ 0, where F(t)=∫0tf(s)ds;
lim inf|t|→0F(t)|t|4N+2>Λ.

When N ≥ 3, by using the Pohoăev identity argument, Wang and Liao [41] obtained the same conclusion only under (PG) and (ZS). When f(u) = λ|u|^p–2u with λ > 0 and 2 < p < 2^*, Tang, Wei and Chen [38] obtained the existence of a ground state solution to (1.2) for every p ∈ (2, 2^*). For more existence results on (1.1) or related results, we refer to [1, 4, 5, 8, 12, 13, 14, 15, 18, 25, 26, 29, 30, 32, 33, 34, 42].

In above-mentioned works [35, 38, 41], it was only considered the case when f(u) has polynomial growth. When N = 2, the corresponding Sobolev embedding yields H¹(ℝ²) ⊂ L^s(ℝ²) for all s ∈ [2, +∞), but H¹(ℝ²) ⊈ L^∞(ℝ²). In this case, the Pohozaev-Trudinger-Moser inequality in ℝ² can be seen as a substitute of the Sobolev inequality, which was first established by Cao in [11], see also [2, 10], and reads as follows.

Lemma 1.1

If β > 0 and u ∈ H¹(ℝ²), then

∫R2eβu2−1dx<∞;
if u ∈ H¹(ℝ²), ∥∇u∥22 ≤ 1, ∥u∥₂ ≤ M < ∞, and β < 4π, then there exists a constant 𝓒(M, β), which depends only on M and β, such that

∫R2eβu2−1dx≤C(M,β).

Based on Lemma 1.1, we say f(t) has subcritical exponential growth at t = ±∞ if it verifies

f ∈ 𝓒(ℝ, ℝ) and

lim|t|→∞|f(t)|eβt2=0,for all β>0; (1.3)

and f(t) has critical exponential growth at t = ±∞ if it verifies
f ∈ 𝓒(ℝ, ℝ) and there exists β₀ > 0 such that

lim|t|→∞|f(t)|eβt2=0,for all β>β0 (1.4)

and

lim|t|→∞|f(t)|eβt2=+∞,for all β<β0. (1.5)

This notion of criticality was introduced by Adimurthi and Yadava [3], see also de Figueiredo, Miyagaki and Ruf [16], for the study of the planar Schrödinger equation

−Δu+V(x)u=f(x,u),x∈R2,u∈H1(R2), (1.6)

which is the maximal growth that allows to treat the problem variationally in H¹(ℝ²).

Inspired by [16, 35, 38, 41], in the present paper, we consider the following planar Choquard equation:

−Δu+u=Iα∗|u|α2+1|u|α2−1u+f(u),x∈R2;u∈H1(R2), (1.7)

where α ∈ (0, 2), f satisfies (F1) or (F1′), and I_α : ℝ² → ℝ is the Riesz potential defined by

Iα(x)=Γ2−α2Γα22απ|x|2−α,x∈R2∖{0}.

In (1.7), as what mentioned before, α2+1 is the lower critical exponent coming from the Hardy-Littlewood-Sobolev inequality. Naturally, we are interested in whether (1.7) admits a nontrivial solution or a ground state solution when f has critical exponential growth in the sense of Trudinger-Moser. The double difficulties due to both the lower critical exponent α2+2 and the critical exponential growth of f enforce the implementation of new ideas and tricks.

To state our theorems, we need to introduce some notations and assumptions.

By the Hardy-Littlewood-Sobolev inequality, one has

S∫R2Iα∗|u|α2+1|u|α2+1dx22+α≤∫R2u2dx. (1.8)

In view of [20, Theorem 4.3], the sharp constant 𝓢 is achieved by a function u ∈ H¹(ℝ²) if and only if for every x ∈ ℝ²,

u(x)=A(y2+|x−a|2)−1 (1.9)

for a ∈ ℝ², A > 0 and y > 0. In what follows, we let A = A₀ and A₀ is determined by

A0α+2∫R2∫R2dxdy(1+|x|2)α2+1|x−y|2−α(1+|y|2)α2+1=2απΓα2Γ1−α2. (1.10)

Set T0=(α+2)(3S+2πA02)61/α. Let t₀ be the unique positive root in the interval (0,S1α) of the following equation

2π(p−2)A0234pp1p−2t2=αS2α+1−(α+2)St2+2tα+2α+2, (1.11)

and let s₀ be the unique positive root in the interval (0,S1α) of the following equation

4π3pp2(p−2)p2−1A0ptp=αα+2S2α+1−4πβ0++2πA023t24πα−(α+2)β0St2+2β0tα+22(α+2)β0p2−1, (1.11)

where a₊ = max{a, 0}.

In addition (F1) and (F1′), we also assume f satisfies the following conditions:

f(t) = o(|t|) as t → 0;
lim inf|t|→0F(t)|t|4>A0−2S−2α;
there exist p > 4 and λ > p−13(A0t0)p−2 such that

F(t)≥λ|t|p,∀0≤t≤A0T0;
there exist p > 2 and λ > λ₀ such that

F(t)≥λ|t|p,∀0≤t≤A0T0,

where

λ0=(p−1)3αβ0S2α+1−4π++2π(α+2)β0(A0s0)26π(α+2)β0(A0s0)p;

(WN) f(t)|t| is non-decreasing on (–∞, 0) ∪ (0, ∞).

In view of (1.8) and Lemma 1.1, under (F1) (or (F1′)) and (F2), the energy functional 𝓘 : H¹(ℝ²) → ℝ associated with (1.7)

I(u)=12∫R2|∇u|2+u2dx−12+α∫R2Iα∗|u|α2+1|u|α2+1dx−∫R2F(u)dx (1.13)

is continuously differentiable, and

〈I′(u),v〉=∫R2∇u⋅∇v+uvdx−∫R2Iα∗|u|α2+1|u|α2−1uvdx−∫R2f(u)vdx,∀u,v∈H1(R2), (1.14)

moreover its critical points correspond to the weak solutions of (1.7). As usual, a solution is called a ground state solution if its energy is minimal among all nontrivial solutions.

Our main results are as follows.

Theorem 1.2

Assume that (F1), (F2), (F3) (or (F4)) and (AR) hold. Then (1.7) has a solution ū ∈ H¹(ℝ²) ∖ {0}.

Theorem 1.3

Assume that (F1), (F2), (F3) (or (F4)) and (WN) hold. Then (1.7) has a solution ū ∈ 𝓝 such that

I(u¯)=infNI=infu∈H1(R2)∖{0}maxt>0I(tu)>0,

where

N:=u∈H1(R2)∖{0}:〈I′(u),u〉=0 (1.15)

is the Nehari manifold of 𝓘.

Theorem 1.4

Assume that (F1′), (F2) and (AR) with μ = α + 2 hold. Further suppose that one of the following conditions:

β₀ ≤ 4πS−2α−1 and (F3) (or (F4)) hold;
β₀ > 4πS−2α−1 and (F5) holds.

Then (1.7) has a solution ū ∈ H¹(ℝ²) ∖ {0}.

Theorem 1.5

Assume that (F1′), (F2) and (WN) hold. Further suppose that one of the following conditions:

β₀ ≤ 4πS−2α−1 and (F3) (or (F4)) hold;
β₀ > 4πS−2α−1 and (F5) holds.

Then (1.7) has a solution ū ∈ 𝓝 such that

I(u¯)=infNI=infu∈H1(R2)∖{0}maxt>0I(tu)>0.

The paper is organized as follows. In Section 2, we give some useful lemmas. We give the proofs of Theorems 1.2 and 1.3 in Sections 3 and prove Theorems 1.4 and 1.5 in Section 4.

Throughout the paper we also make use of the following notations:

H¹(ℝ²) denotes the usual Sobolev space equipped with the inner product and norm

(u,v)=∫R2(∇u⋅∇v+uv)dx,∥u∥=(u,u)1/2,∀u,v∈H1(R2).
L^s(ℝ²) (1 ≤ s < ∞) denotes the Lebesgue space with the norm ∥u∥_s = (∫_ℝ²|u|^s dx)^1/s.
C₁, C₂, ⋯ denote positive constants possibly different in different places.

2 Some useful lemmas

By a simple calculation, we can verify the following lemma.

Lemma 2.1

Assume that (F1) (or (F1′)), (F2) and (WN) hold. Then the following two inequalities hold:

1−t22f(τ)τ−F(τ)+F(tτ)≥0,∀t∈[0,+∞) (2.1)

and

α−(α+2)t2+2tα+2>0,∀t∈[0,1)∪(1,+∞). (2.2)

Inspired by [37], we establish a key functional inequality as follows.

Lemma 2.2

Assume that (F1) (or (F1′)), (F2) and (WN) hold. Then for any u ∈ H¹(ℝ²) and t ≥ 0, one has

I(u)≥I(tu)+1−t22〈I′(u),u〉. (2.3)

Proof

Note that

I(tu)=t22∥u∥2−tα+2α+2∫R2Iα∗|u|α2+1|u|α2+1dx−∫R2F(tu)dx. (2.4)

Thus, by (1.13), (1.14), (2.1), (2.2) and (2.4), one has

I(u)−I(tu) = 1−t22∥u∥2−1−tα+2α+2∫R2Iα∗|u|α2+1|u|α2+1dx−∫R2[F(u)−F(tu)]dx=1−t22∥u∥2−∫R2Iα∗|u|α2+1|u|α2+1dx−∫R2f(u)udx+∫R21−t22f(u)u−F(u)+F(tu)dx+α−(α+2)t2+2tα+22(α+2)∫R2Iα∗|u|α2+1|u|α2+1dx≥1−t22〈I′(u),u〉.

This shows that (2.3) holds.□

From Lemma 2.2, we have the following corollary.

Corollary 2.3

Assume that (F1) (or (F1′)), (F2) and (WN) hold. Then for u ∈ 𝓝,

I(u)=maxt>0I(tu). (2.5)

Lemma 2.4

Assume that (F1) (or (F1′)) and (F2) hold. Then there exists ρ₀ > 0 such that

κ:={I(u):u∈H1(R2),∥u∥=ρ0}>0. (2.6)

Proof

By (F1) (or (F1′)) and (F2), one has for some constants α > 0 and C₁ > 0

|F(t)|≤14t2+C1eαt2−1t3,∀(x,t)∈R2×R. (2.7)

In view of Lemma 1.1 ii), we have

∫R2e2αu2−1dx=∫R2e2α∥u∥2(u/∥u∥)2−1dx≤C(1,2π),∀∥u∥≤π/α. (2.8)

From (2.7) and (2.8), we obtain

∫R2F(u)dx ≤ 14∥u∥22+C1∫R2eαu2−1|u|3dx≤14∥u∥22+C1∫R2e2αu2−1dx1/2∥u∥63≤14∥u∥2+C2∥u∥3,∀∥u∥≤π/α. (2.9)

Hence, it follows from (1.8), (1.13) and (2.9) that

I(u) = 12∥u∥2−1α+2∫R2Iα∗|u|α2+1|u|α2+1dx−∫R2F(u)dx≥14∥u∥2−C2∥u∥3−C3∥u∥α+2,∀∥u∥≤π/α.

Therefore, there exists 0<ρ0<π/α such that (2.6) holds.□

By Lemma 2.4 and the classical mountain pass theorem [40], we can prove the following lemma by a standard argument.

Lemma 2.5

Assume that (F1) (or (F1′)) and (F2) hold. If F(t) ≥ 0, then there exists a sequence {u_n} ⊂ H¹(ℝ²) satisfying

I(un)→c0,∥I′(un)∥(1+∥un∥)→0. (2.10)

where

c0=infy∈Γmaxt∈[0,1]I(y(t)) (2.11)

and

Γ={y∈C([0,1],H1(R2)):y(0)=0,I(y(1))<0}. (2.12)

Lemma 2.6

Assume that (F1) (or (F1′)), (F2) and (WN) hold. Then for any u ∈ H¹(ℝ²) ∖ {0}, there exists a t_u > 0 such that t_uu ∈ 𝓝.

Proof

Let u ∈ H¹(ℝ²) ∖ {0} be fixed and define a function ζ(t) := 𝓘(tu) on [0, ∞). Clearly, by (1.14) and (2.4), we have

ζ′(t)=0⇔t2∥u∥2−tα+2∫R2Iα∗|u|α2+1|u|α2+1dx−∫R2f(tu)tudx=0⇔〈I′(tu),tu〉=0⇔tu∈N. (2.13)

By Lemma 2.4, one has ζ(0) = 0 and ζ(t) > 0 for t > 0 small and ζ(t) < 0 for t large. Therefore max_t∈(0,∞) ζ(t) is achieved at some t_u > 0 so that ζ′(t_u) = 0 and t_uu ∈ 𝓝.□

From Corollary 2.3 and Lemma 2.6, we have 𝓝 ≠ ∅ and the following minimax characterization.

Lemma 2.7

Assume that (F1) (or (F1′)), (F2) and (WN) hold. Then inf_u∈𝓝 𝓘(u) := m = inf_{u∈H¹(ℝ²)∖{0}} max_t>0𝓘(tu).

By Corollary 2.3 and Lemma 2.6, we have the following lemma.

Lemma 2.8

Assume that (F1) (or (F1′)), (F2) and (WN) hold. Then

m≥κ={I(u):u∈H1(R2),∥u∥=ρ0}>0,

where κ and ρ₀ are given in Lemma 2.4.

Next, we apply the non-Nehari manifold method introduced in [36] to prove the following lemma which is key to obtain a Nehari-type ground state solution.

Lemma 2.9

Assume that (F1) (or (F1′)), (F2) and (WN) hold. Then there exist a constant c_* ∈ [κ, m] and a sequence {u_n} ⊂ H¹(ℝ²) satisfying

I(un)→c∗,∥I′(un)∥(1+∥un∥)→0. (2.14)

Proof

Choose v_k ∈ 𝓝 such that

m≤I(vk)<m+1k,k∈N. (2.15)

From (WN) and (2.4), we have

I(tvk) = t22∥vk∥2−tα+2α+2∫R2Iα∗|vk|α2+1|vk|α2+1dx−∫R2F(tvk)dx≤t22∥vk∥2−tα+2α+2∫R2Iα∗|vk|α2+1|vk|α2+1dx. (2.16)

It follows that there exists t_k > 0 such that 𝓘(t_kv_k) < 0 for all k ∈ ℕ. Hence, in view of Lemma 2.5, there exist a constant c_k ∈ [κ, sup_{t∈[0,t_k]} 𝓘(tv_k)] and a sequence {u_k,n}_n∈ℕ ⊂ H¹(ℝ²) satisfying

I(uk,n)→ck,∥I′(uk,n)∥(1+∥uk,n∥)→0,k∈N. (2.17)

By virtue of Corollary 2.4, one can get that

I(vk)≥I(tvk),∀t≥0. (2.18)

Hence, by (2.15), (2.17) and (2.18), one has

I(uk,n)→ck<m+1k,∥I′(uk,n)∥(1+∥uk,n∥)→0,k∈N.

Now, we can choose a sequence {n_k} ⊂ ℕ such that

I(uk,nk)<m+1k,∥I′(uk,nk)∥(1+∥uk,nk∥)<1k,k∈N.

Let u_k = u_{k,n_k}, k ∈ ℕ. Then, going if necessary to a subsequence, we have

I(un)→c∗∈[κ,m],∥I′(un)∥(1+∥un∥)→0.

□

Next, we give an estimate on the energy level m, which is essential in ensuring compactness.

Lemma 2.10

Assume that (F1) (or (F1′)), (F2) and (F3) hold, moreover F(t) ≥ 0 for all t ∈ ℝ. Then max{c₀, m} < m∗:=α2(α+2)S2α+1.

Proof

We set U(x) = A₀(1 + |x|²)^–1, where A₀ satisfies (1.10). By a simple calculation, we have

∥U∥44=∫R2|U|4dx=2πA04∫0+∞r(1+r2)4dr=πA043 (2.19)

and

∥∇U∥22=∫R2|∇U|2dx=8πA02∫0+∞r3(1+r2)4dr=2πA023. (2.20)

By (F3), we can choose Λ₀ > 0 such that

lim inf|t|→0F(t)|t|4≥Λ0>A0−2S−2α. (2.21)

Let t∗=S1α. Then we can choose ϵ > 0 such that

(Λ0−ϵ)(t∗−ϵ)21+ϵ≥∥∇U∥222∥U∥44. (2.22)

We set

g(t)=S2t2−tα+2α+2. (2.23)

By (2.21), we can choose ε ∈ (0, 1) such that

F(εtU(x))|εtU(x)|4≥Λ0−ϵ,∀x∈R2,0≤t≤T0 (2.24)

and

12ε2T02∥∇U∥22<m∗−g(t∗+ϵ),12ε2t∗2∥∇U∥22<m∗−g(t∗−ϵ), (2.25)

where

T0=(α+2)(3S+2πA02)61α=(α+2)(S+∥∇U∥22)21α. (2.26)

Now we define a function φ_ε(t) as follows:

φε(t)=ε2∥∇U∥222t2−ε−2∫R2F(εtU)dx. (2.27)

It is easy to see that g(t)<g(t∗)=α2(α+2)S2α+1=m∗ for t ∈ [0, t_*) ∪ (t_*, ∞). We set U_ε(x) = εU(εx). Then it follows from the definition of 𝓢 that

∥Uε∥22=Sand∫R2Iα∗|Uε|α2+1|Uε|α2+1dx=1 (2.28)

and

∥∇Uε∥22=ε2∥∇U∥22. (2.29)

From (2.4), (2.23), (2.27), (2.28) and (2.29), we have

I(tUε) = 12∥∇Uε∥22+∥Uε∥22t2−tα+2α+2∫R2Iα∗|Uε|α2+1|Uε|α2+1dx−∫R2F(tUε)dx=S2t2−tα+2α+2+ε2∥∇U∥222t2−ε−2∫R2F(tεU)dx=g(t)+φε(t). (2.30)

There are four possible subcases:

Subcase i) t ≥ T₀. Then it follows from (2.23), (2.26), (2.27) and (2.30) that

maxt≥T0I(tUε)=maxt≥T0g(t)+φε(t)≤0<m∗.

Subcase ii) t_* + ϵ ≤ t ≤ T₀. Then it follows from (2.23), (2.25), (2.27) and (2.30) that

maxt∗+ϵ≤t≤T0I(tUε) = maxt∗+ϵ≤t≤T0g(t)+φε(t)≤g(t∗+ϵ)+12ε2T02∥∇U∥22<m∗.

Subcase iii) t_* – ϵ ≤ t ≤ t_* + ϵ. Then from (2.22), (2.24) and (2.27), one has

φε(t) = ε2∥∇U∥222t2−ε−2∫R2F(εtU)dx≤ε2∥∇U∥222t2−(Λ0−ϵ)ε2∥U∥44t4≤−ϵε2∥∇U∥222(t∗−ϵ)2. (2.31)

Hence, it follows from (2.23), (2.27), (2.30) and (2.31) that

maxt∗−ϵ≤t≤t∗+ϵI(tUε) = maxt∗−ϵ≤t≤t∗+ϵg(t)+φε(t)≤g(t∗)+maxt∗−ϵ≤t≤t∗+ϵφε(t)≤m∗−12ϵε2(t∗−ϵ)2∥∇U∥22<m∗.

Subcase iv) 0 ≤ t ≤ t_* – ϵ. Then it follows from (2.23), (2.25), (2.27) and (2.30) that

max0≤t≤t∗−ϵI(tUε) = max0≤t≤t∗−ϵg(t)+φε(t)≤g(t∗−ϵ)+12ε2t∗2∥∇U∥22<m∗.

The above four subcases and Lemmas 2.5 and 2.7 show that

max{c0,m}≤maxt≥0I(tUε)<m∗.

□

Lemma 2.11

Assume that (F1) (or (F1′)), (F2) and (F4) hold, moreover F(t) ≥ 0 for all t ∈ ℝ. Then max{c₀, m} < m∗=α2(α+2)S2α+1.

Proof

As in the proof of Lemma 2.10, we set U(x) = A₀(1 + |x|²)^–1. By a simple calculation, we have

∥U∥pp = ∫R2|U|pdx=2πA0p∫0+∞r(1+r2)pdr=πA0p∫0+∞dt(1+t)p=πA0pp−1. (2.32)

Now we define a function φ(t) as follows:

φ(t)=∥∇U∥222t2−∫R2F(tU)dx. (2.33)

It follows from the definition of 𝓢 that

∥U∥22=Sand∫R2Iα∗|U|α2+1|U|α2+1dx=1. (2.34)

From (2.4), (2.23), (2.33) and (2.34), we have

I(tU) = 12∥∇U∥22+∥U∥22t2−tα+2α+2∫R2Iα∗|U|α2+1|U|α2+1dx−∫R2F(tU)dx=S2t2−tα+2α+2+∥∇U∥222t2−∫R2F(tU)dx=g(t)+φ(t). (2.35)

By (F4), (1.11), (2.20) and (2.32), we have

λ>∥∇U∥222∥U∥ppt0p−2 (2.36)

and

λ>p−22pp−22∥∇U∥2p[m∗−g(t0)](p−2)/2p∥U∥pp. (2.37)

From (F4) and (2.33), one has

φ(t)≤∥∇U∥222t2−λ∥U∥pptp,∀0≤t≤T0. (2.38)

There are three possible subcases:

Subcase i) t ≥ T₀. Then it follows from (2.23), (2.33) and (2.35) that

maxt≥T0I(tU)=maxt≥T0g(t)+φ(t)≤0<m∗.

Subcase ii) t₀ ≤ t ≤ T₀. Then it follows from (2.23), (2.35), (2.36) and (2.38) that

maxt0≤t≤T0I(tU) = maxt0≤t≤T0g(t)+φ(t)≤g(t∗)+maxt0≤t≤T0φ(t)≤m∗+12t02∥∇U∥22−2λ∥U∥ppt0p−2<m∗.

Subcase iii) 0 ≤ t ≤ t₀. Then it follows from (2.23), (2.35), (2.37) and (2.38) that

max0≤t≤t0I(tU) = max0≤t≤t0g(t)+φ(t)≤g(t0)+(p−2)∥∇U∥22p/(p−2)2p(pλ∥U∥pp)2/(p−2)<m∗.

The above three subcases and Lemmas 2.5 and 2.7 show that

max{c0,m}≤maxt≥0I(tUε)<m∗.

□

Lemma 2.12

Assume that (F1′), (F2) and (F5) hold, moreover F(t) ≥ 0 for all t ∈ ℝ. Then max{c₀, m} < ω₀ := 2πα(α+2)β0.

Proof

We set U(x), g(t) and φ(t) are the same as in in the proof of Lemma 2.11. Then (2.20), (2.26), (2.32) and (2.35) hold. By (F5), (1.11), (2.20) and (2.32), we have

λ>2(m∗−ω0)++∥∇U∥22s022∥U∥pps0p (2.39)

and

λ>p−22pp−22∥∇U∥2p[ω0−g(s0)](p−2)/2p∥U∥pp. (2.40)

From (F5) and (2.33), one has

φ(t)≤∥∇U∥222t2−λ∥U∥pptp,∀0≤t≤T0. (2.41)

There are three possible subcases:

Subcase i) t ≥ T₀. Then it follows from (2.23), (2.26), (2.33) and (2.35) that

maxt≥T0I(tU) = maxt≥T0g(t)+φ(t)≤0<ω0.

Subcase ii) s₀ ≤ t ≤ T₀. Then it follows from (2.23), (2.35), (2.39) and (2.41) that

maxs0≤t≤T0I(tU) = maxs0≤t≤T0g(t)+φ(t)≤g(t∗)+maxs0≤t≤T0φ(t)≤m∗+12s02∥∇U∥22−2λ∥U∥pps0p−2<ω0.

Subcase iii) 0 ≤ t ≤ s₀. Then it follows from (2.23), (2.35), (2.40) and (2.41) that

max0≤t≤s0I(tU) = max0≤t≤s0g(t)+φ(t)≤g(s0)+(p−2)∥∇U∥22p/(p−2)2p(pλ∥U∥pp)2/(p−2)<ω0.

The above three subcases and Lemmas 2.5 and 2.7 show that

max{c0,m}≤maxt≥0I(tUε)<ω0.

□

3 Sub-critical case

Lemma 3.1

Assume that (F1), (F2) and (AR) hold. Then any sequence {u_n} ⊂ H¹(ℝ²) satisfying (2.10) is bounded in H¹(ℝ²).

The proof of Lemma 3.1 is standard, so we omit it.

Lemma 3.2

Assume that (F1), (F2), (F3) (or (F4)) and (WN) hold. Then any sequence {u_n} ⊂ H¹(ℝ²) satisfying (2.14) is bounded in H¹(ℝ²).

Proof

To prove the boundedness of {u_n}, arguing by contradiction, suppose that ∥u_n∥ → ∞. Let v_n = u_n/∥u_n∥. Then ∥v_n∥ = 1. Passing to a subsequence, we may assume that v_n ⇀ v in H¹(ℝ²), v_n → v in Llocs (ℝ²), 2 ≤ s < ∞, v_n → v a.e. on ℝ². If

δ:=lim supn→∞supy∈R2∫B1(y)vn2dx=0,

then by Lions’ concentration compactness principle [40, Lemma 1.21], v_n → 0 in L^s(ℝ²) for 2 < s < ∞. Set β∈(0,1/St∗2), where 𝓢 is determined by (1.8) and t_* is given in the proof of Lemma 2.10. In view of Lemmas 2.10 and 2.11, we have m < m_*. By (F1) and (F2), there exists C₁ > 0 such that

|F(t)|≤m∗−m4t∗2St2+C1|t|eβt2−1,∀t∈R. (3.1)

Then (3.1) and Lemma 1.1 ii) lead to

∫R2F(t∗Svn)dx ≤ m∗−m4∥vn∥22+C1t∗S∫R2eβSt∗2vn2−1|vn|dx≤m∗−m4+C1t∗S∫R2eβSt∗2vn2−13/2dx2/3∥vn∥3≤m∗−m4+C1t∗S∫R2e3βSt∗2vn2/2−1dx2/3∥vn∥3≤m∗−m4+o(1). (3.2)

Let tn=t∗S/∥un∥. Hence, from (1.8), (2.3), (2.4), (2.14) and (3.2), we derive

c∗+o(1) = I(un)≥I(tnun)+1−tn22〈I′(un),un〉=tn22∥un∥2−tnα+2α+2∫R2Iα∗|un|α2+1|un|α2+1dx−∫R2F(tnun)dx+o(1)=S2t∗2−Sα2+1t∗α+2α+2∫R2Iα∗|vn|α2+1|vn|α2+1dx−∫R2F(t∗Svn)dx+o(1)≥S2t∗2−t∗α+2α+2∥vn∥2α+2−m∗−m4+o(1)≥S2t∗2−t∗α+2α+2−m∗−m4+o(1)=m∗−m∗−m4+o(1),

which is a contradiction due the fact that c^* ≤ m < m_*. This shows that δ > 0. The rest of the proof is standard, so we omit it.□

Lemma 3.3

Assume that (F1) (or (F1′)) and (F2) hold. Let v_n ⇀ v̄ in H¹(ℝ²) and

∫R2f(vn)vndx≤K0 (3.3)

for some constant K₀ > 0. Then for every ϕ ∈ C0∞ (ℝ²)

limn→∞∫R2f(vn)ϕdx=∫R2f(v¯)ϕdx. (3.4)

Proof

Let Ω = supp ϕ. For any given ε > 0, we have

∫|vn|≥K0∥ϕ∥∞ε−1|f(vn)ϕ|dx≤εK0∫|vn|≥K0∥ϕ∥∞ε−1f(vn)vndx<ε. (3.5)

Since f(v̄)ϕ ∈ L¹(Ω), it follows that there exists δ > 0 such that

∫A|f(v¯)ϕ|dx<εifmeas(A)≤δ (3.6)

for all measurable set A ⊂ Ω. Next using the fact that v̄ ∈ L¹(Ω) we find M₁ > 0 such that

meas({x∈Ω:|v¯(x)|≥M1})≤δ. (3.7)

Let M_ε = max{M₁, K₀∥ϕ∥_∞ε^–1}. Then we have

∫|vn|≥Mε|f(vn)ϕ|dx<ε,∫|v¯|≥Mε|f(x,v¯)ϕ|dx<ε. (3.8)

Since |f(v_n)ϕ|χ_{|v_n|≤M_ε} → |f(v̄)ϕ|χ_{|v̄|≤M_ε} a.e. in Ω, moreover,

|f(vn)ϕ|χ|vn|≤Mε≤∥ϕ∥∞max|t|≤Mε|f(t)|<∞,∀x∈Ω.

Then Lebesgue dominated convergence theorem leads to

∫|vn|≤Mε|f(vn)ϕ|dx→∫|v¯|≤Mε|f(v¯)ϕ|dx. (3.9)

It follows from (3.8) and (3.9) that (3.4) holds due to the arbitrariness of ε > 0.□

Proof of Theorem 1.2

By Lemmas 2.5, 2.10, 2.11 and 3.1, there exists a bounded sequence {u_n} ⊂ H¹(ℝ²) satisfying (2.10) with c₀ ∈ [κ, m_*). Choose C₂ > 0 such that ∥u_n∥ ≤ C₂. If

δ:=lim supn→∞supy∈R2∫B1(y)un2dx=0,

then by Lions’ concentration compactness principle [40, Lemma 1.21], one has u_n → 0 in L^s(ℝ²) for 2 < s < ∞. Let β ∈ (0, 1/ C22 ). By virtue of (F1) and (F2), for any ε > 0, there exists C_ε > 0 such that

|f(t)|≤ε|t|+Cεeβt2−1,∀t∈R. (3.10)

From (3.10) and Lemma 1.1 ii), similarly to (3.2), we can prove that

∫R2f(un)undx ≤ ε∥un∥22+Cε∫R2eβun2−1|un|dx≤C22ε+o(1). (3.11)

Due to the arbitrariness of ε > 0, we obtain from (3.11)

∫R2f(un)undx=o(1). (3.12)

Similarly, we have

∫R2F(un)dx=o(1). (3.13)

From (1.13), (1.14), (2.10), (3.12) and (3.13), one can get

∥un∥2=∫R2Iα∗|un|α2+1|un|α2+1dx+o(1) (3.14)

and

c0+o(1) = I(un)=12∥un∥2−1α+2∫R2Iα∗|un|α2+1|un|α2+1dx+o(1)=α2(α+2)∫R2Iα∗|un|α2+1|un|α2+1dx+o(1). (3.15)

It follows from (1.8), (3.14) and (3.15) that

c0+o(1) = α2(α+2)∫R2Iα∗|un|α2+1|un|α2+1dx+o(1)≥α2(α+2)∥un∥22+o(1)≥αS2(α+2)∫R2Iα∗|un|α2+1|un|α2+1dx2α+2+o(1)=αS2(α+2)2c0(α+2)α2α+2+o(1), (3.16)

which leads to

c0≥α2(α+2)S2α+1=m∗. (3.17)

This contradicts with the fact that c₀ < m_*. Hence, δ > 0, and so there exists a sequence {y_n} ⊂ ℝ² such that ∫_{B₁(y_n)}|u_n|² dx > δ/2 > 0. Let û_n(x) = u_n(x + y_n). Then we have ∥û_n∥ = ∥u_n∥ and

I(u^n)→c0,∥I′(u^n)∥(1+∥u^n∥)→0,∫B1(0)|u^n|2dx>δ2. (3.18)

Therefore, there exists û ∈ H¹(ℝ²) ∖ {0} such that, passing to a subsequence,

u^n⇀u^,inH1(R2);u^n→u^,inLlocs(R2),∀s∈[1,∞);u^n→u^,a.e. onR2. (3.19)

From (1.14), (3.18) and (3.19), we have

C22≥∥u^n∥2=∫R2Iα∗|u^n|α2+1|u^n|α2+1dx+∫R2f(u^n)u^ndx+o(1) (3.20)

Therefore, (1.14), (3.20) and Lemma 3.3 yield for every ϕ ∈ C0∞ (ℝ²),

〈I′(u^),ϕ〉=limn→∞〈I′(u^n),ϕ〉=0.

Hence 𝓘′(û) = 0. This completes the proof.□

Proof of Theorem 1.3

By Lemmas 2.9, 2.10, 2.11 and 3.2, there exists a bounded sequence {u_n} ⊂ H¹(ℝ²) satisfying (2.14) with c_* ∈ [κ, m_*). Similarly to the proof of Theorem 1.2, we can prove that there exists a sequence {û_n} and û ∈ H¹(ℝ²) ∖ {0} such that

u^n⇀u^,inH1(R2);u^n→u^,inLlocs(R2),∀s∈[1,∞);u^n→u^,a.e. onR2 (3.21)

and

I(u^n)→c∗,∥I′(u^n)∥(1+∥u^n∥)→0,I′(u^)=0. (3.22)

It follows that 𝓘(û) ≥ m. From (WN), (1.13), (1.14), (3.21), (3.22) and Fatou’ Lemma, we have

m ≥ c∗=limn→∞I(u^n)−12〈I′(u^n),u^n〉=limn→∞α2(α+2)∫R2Iα∗|u^n|α2+1|u^n|α2+1dx+12∫R2[f(u^n)u^n−2F(u^n)]dx≥α2(α+2)∫R2Iα∗|u^|α2+1|u^|α2+1dx+12∫R2[f(u^)u^−2F(u^)]dx=I(u^)−12〈I′(u^),u^〉=I(u^)≥m.

This shows that 𝓘(û) = m, which, together with Lemma 2.7, completes the proof.□

4 Critical case

Lemma 4.1

Assume that (F1′), (F2) and (AR) with μ = α + 2 hold. Then any sequence {u_n} ⊂ H¹(ℝ²) satisfying (2.10) is bounded in H¹(ℝ²).

The proof of Lemma 4.1 is standard, so we omit it.

Lemma 4.2

Suppose all assumptions in Theorem 1.5 hold. Then any sequence {u_n} ⊂ H¹(ℝ²) satisfying (2.14) is bounded in H¹(ℝ²).

Proof

δ:=lim supn→∞supy∈R2∫B1(y)vn2dx=0,

then by Lions’ concentration compactness principle [40, Lemma 1.21], v_n → 0 in L^s(ℝ²) for 2 < s < ∞. There are two cases to distinguish:

Case 1). β₀ ≤ 4πS−2α−1 and (F3) (or (F4)) hold. In view of Lemmas 2.10 and 2.11, we have c_* ≤ m < m_*. Let c_* := (1 – 3ϵ)m_*. By (F1′) and (F2), there exists C₁ > 0 such that

|F(t)|≤ϵm∗t∗2(1−ϵ)St2+C1|t|e(1+ϵ)β0t2−1,∀t∈R. (4.1)

Let q ∈ (1, 2) such that q(1 – ϵ²) < 1 and let q′ = q/(q – 1). Then (4.1) and Lemma 1.1 ii) lead to

∫R2F(t∗(1−ϵ)Svn)dx ≤ ϵm∗∥vn∥22+C1t∗S∫R2e(1−ϵ2)β0St∗2vn2−1|vn|dx≤ϵm∗+C1t∗S∫R2e(1−ϵ2)β0St∗2vn2−1qdx1/q∥vn∥q′≤ϵm∗+C1t∗S∫R2eq(1−ϵ2)β0St∗2vn2−1dx1/q∥vn∥q′≤ϵm∗+C1t∗S∫R2e4πq(1−ϵ2)vn2−1dx1/q∥vn∥q′≤ϵm∗+o(1). (4.2)

Let tn=t∗(1−ϵ)S/∥un∥. Hence, from (1.8), (2.3), (2.4), (2.14) and (4.2), we derive

c∗+o(1) = I(un)≥I(tnun)+1−tn22〈I′(un),un〉=tn22∥un∥2−tnα+2α+2∫R2Iα∗|un|α2+1|un|α2+1dx−∫R2F(tnun)dx+o(1)=(1−ϵ)S2t∗2−[(1−ϵ)S]α2+1t∗α+2α+2∫R2Iα∗|vn|α2+1|vn|α2+1dx−∫R2F(t∗(1−ϵ)Svn)dx+o(1)≥(1−ϵ)S2t∗2−(1−ϵ)α2+1α+2∥vn∥2α+2t∗α+2−ϵm∗+o(1)≥(1−ϵ)S2t∗2−t∗α+2α+2−ϵm∗+o(1)=(1−ϵ)m∗−ϵm∗+o(1),

which is a contradiction.

Case 2). β₀ > 4πS−2α−1 and (F5) holds. In view of Lemma 2.12, we deduce that c_* ≤ m < ω₀ < m_*. Let c_* := (1 – 3ϵ)ω₀ and t∗∗=4πβ0S. It is easy to verify t_** ∈ (0, t_*) and

S2t∗∗2−1α+2t∗∗α+2>2πα(α+2)β0=ω0. (4.3)

By (F1′) and (F2), there exists C₂ > 0 such that

|F(t)|≤ϵω0t∗∗2(1−ϵ)St2+C2|t|e(1+ϵ)β0t2−1,∀t∈R. (4.4)

Let q ∈ (1, 2) such that q(1 – ϵ²) < 1 and let q′ = q/(q – 1). Then (4.4) and Lemma 1.1 ii) lead to

∫R2F(t∗∗(1−ϵ)Svn)dx ≤ ϵω0∥vn∥22+C2t∗∗S∫R2e(1−ϵ2)β0St∗∗2vn2−1|vn|dx≤ϵω0+C2t∗∗S∫R2e(1−ϵ2)β0St∗∗2vn2−1qdx1/q∥vn∥q′≤ϵω0+C2t∗∗S∫R2eq(1−ϵ2)β0St∗∗2vn2−1dx1/q∥vn∥q′=ϵω0+C2t∗∗S∫R2e4πq(1−ϵ2)vn2−1dx1/q∥vn∥q′≤ϵω0+o(1). (4.5)

Let tn=t∗∗(1−ϵ)S/∥un∥. Hence, from (1.8), (2.3), (2.4), (2.14), (4.3) and (4.5), we derive

c∗+o(1) = I(un)≥I(tnun)+1−tn22〈I′(un),un〉=tn22∥un∥2−tnα+2α+2∫R2Iα∗|un|α2+1|un|α2+1dx−∫R2F(tnun)dx+o(1)=(1−ϵ)S2t∗∗2−[(1−ϵ)S]α2+1t∗∗α+2α+2∫R2Iα∗|vn|α2+1|vn|α2+1dx−∫R2F(t∗∗(1−ϵ)Svn)dx+o(1)≥(1−ϵ)S2t∗∗2−(1−ϵ)α2+1α+2∥vn∥2α+2t∗∗α+2−ϵω0+o(1)≥(1−ϵ)S2t∗∗2−t∗∗α+2α+2−ϵω0+o(1)>(1−ϵ)ω0−ϵω0+o(1),

which is a contradiction.

Both Case 1) and Case 2) shows that δ > 0. The rest of the proof is standard, so we omit it.□

Proof of Theorem 1.4

By Lemmas 2.5, there exists a sequence {u_n} ⊂ H¹(ℝ²) satisfying (2.10). Since μ = α + 2. Then it follows from (AR), (1.13), (1.14) and (2.10) that

c0+o(1) = I(un)−1α+2〈I′(un),un〉=α2(α+2)∥un∥2+1α+2∫R2[f(un)un−(α+2)F(un)]dx≥α2(α+2)∥un∥2. (4.6)

Hence, it follows from (4.6) that

∥un∥2≤2(α+2)c0α+o(1). (4.7)

There are two cases to distinguish:

Case 1). β₀ ≤ 4πS−2α−1 and (F3) (or (F4)) hold. In view of Lemmas 2.10 and 2.11, we have

2(α+2)c0α<S2α+1≤4πβ0. (4.8)

Case 2). β₀ > 4πS−2α−1 and (F5) holds. In view of Lemma 2.12, we can also deduce

2(α+2)c0α<4πβ0. (4.9)

The above two cases show that there exist ϵ > 0 and q ∈ (1, 2) such that

2(α+2)c0α≤4πβ0(1−3ϵ) (4.10)

and

(1+ϵ)(1−3ϵ)q1−ϵ<1. (4.11)

From (4.7), (4.10) and (4.11), we have

qβ0(1+ϵ)∥un∥2≤1−ϵ+o(1). (4.12)

δ:=lim supn→∞supy∈R2∫B1(y)|un|2dx=0,

then by Lions’ concentration compactness principle [40, Lemma 1.21], one has u_n → 0 in L^s(ℝ²) for 2 < s < ∞. By virtue of (F1′) and (F2), for any ε > 0, there exists C_ε > 0 such that

|f(t)|≤ε|t|+Cεeβ0(1+ϵ)t2−1,∀t∈R. (4.13)

Let q′ = q/(q – 1). It follows from (4.12), (4.13) and Lemma 1.1 ii) that

∫R2f(un)undx ≤ ε∥un∥22+Cε∫R2eβ0(1+ϵ)un2−1|un|dx≤C3ε+Cε∫R2eβ0(1+ϵ)un2−1qdx1/q∥un∥q′≤C3ε+Cε∫R2eqβ0(1+ϵ)un2−1dx1/q∥un∥q′≤C3ε+o(1). (4.14)

Due to the arbitrariness of ε > 0, we obtain from (4.14)

∫R2f(un)undx=o(1). (4.15)

Similarly, we have

∫R2F(un)dx=o(1). (4.16)

The rest proof is the same as one of Theorem 1.2, so we omit it.□

Proof of Theorem 1.5

By Lemmas 2.9 and 4.2, there exists a bounded sequence {u_n} ⊂ H¹(ℝ²) satisfying (2.14). It follows from (WN), (1.13) and (2.14) that 2c_* ≤ lim inf_n→∞ ∥u_n∥² ≤ lim sup_n→∞ ∥u_n∥² < ∞. If

δ:=lim supn→∞supy∈R2∫B1(y)|un|2dx=0,

then by Lions’ concentration compactness principle [40, Lemma 1.21], one has u_n → 0 in L^s(ℝ²) for 2 < s < ∞. It follows that v_n := u_n/∥u_n∥ → 0 in L^s(ℝ²) for 2 < s < ∞. Similarly to the proof of Lemma 4.2, we can show that δ > 0 by distinguishing two cases:

Case 1). β₀ ≤ 4πS−2α−1 and (F3) (or (F4)) hold.

Case 2). β₀ > 4πS−2α−1 and (F5) holds.

Hence, there exists a sequence {y_n} ⊂ ℝ² such that ∫_{B₁(y_n)}|u_n|² dx > δ/2 > 0. Let û_n(x) = u_n(x + y_n). Then we have ∥û_n∥ = ∥u_n∥ and

I(u^n)→c∗,∥I′(u^n)∥(1+∥u^n∥)→0,∫B1(0)|u^n|2dx>δ2. (4.17)

Therefore, there exists û ∈ H¹(ℝ²) ∖ {0} such that, passing to a subsequence,

u^n⇀u^,inH1(R2);u^n→u^,inLlocs(R2),∀s∈[1,2∗);u^n→u^,a.e. onR2. (4.18)

Similarly to the proof of Theorem 1.3, we can prove that 𝓘′(û) = 0 and 𝓘(û) = m.□

Acknowledgement

This work was supported by the NNSF (11971485).

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Received: 2020-03-16

Accepted: 2020-04-21

Published Online: 2020-05-30

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

Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

Abstract

1 Introduction

Lemma 1.1

Theorem 1.2

Theorem 1.3

Theorem 1.4

Theorem 1.5

2 Some useful lemmas

Lemma 2.1

Lemma 2.2

Proof

Corollary 2.3

Lemma 2.4

Proof

Lemma 2.5

Lemma 2.6

Proof

Lemma 2.7

Lemma 2.8

Lemma 2.9

Proof

Lemma 2.10

Proof

Lemma 2.11

Proof

Lemma 2.12

Proof

3 Sub-critical case

Lemma 3.1

Lemma 3.2

Proof

Lemma 3.3

Proof

Proof of Theorem 1.2

Proof of Theorem 1.3

4 Critical case

Lemma 4.1

Lemma 4.2

Proof

Proof of Theorem 1.4

Proof of Theorem 1.5

Acknowledgement

References

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