Multiple solutions for a quasilinear Choquard equation with critical nonlinearity

Rui Li; Yueqiang Song

doi:10.1515/math-2021-0125

Article Open Access

Multiple solutions for a quasilinear Choquard equation with critical nonlinearity

Rui Li and Yueqiang Song

Published/Copyright: December 31, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

In the present work, we are concerned with the multiple solutions for quasilinear Choquard equation with critical nonlinearity in R N . We show multiplicity results for this problem, which are characterized, respectively, by the new version of symmetric mountain-pass theorem and the mountain-pass theorem for even functionals. The novelty of our work is the appearance of the convolution terms as well as critical nonlinearities.

Keywords: quasilinear Choquard equation; Hardy-Littlewood-Sobolev critical exponent; Mountain-pass theorem; concentration-compactness principle

MSC 2010: 35A15; 35J60; 35J20

1 Introduction

In the present paper, we are interested in the existence of multiple solutions for the following quasilinear Choquard equation with critical nonlinearity in R N :

(1.1) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u − a [ Δ ( u 2 ) ] u = α k ( x ) ∣ u ∣ p − 2 u + β ∫ R N ∣ u ∣ 2 2 μ ∗ ∣ x − y ∣ μ d y ∣ u ∣ 2 2 μ ∗ − 2 u , x ∈ R N ,

where a > 0 , b ≥ 0 , 0 < μ < 4 , 1 < p ≤ 4 , N ≥ 3 , α , and β are real parameters, 2 μ ∗ = 2 N − μ N − 2 is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, and k ( x ) ∈ L r ( R N ) with r = 2 2 ∗ 2 2 ∗ − p .

First, we make a quick overview of the literature. To begin with, we note that the following Choquard equation

(1.2) − Δ u + V ( x ) u = ( ∣ x ∣ − μ ∗ F ( u ) ) f ( u ) x ∈ R N

was introduced by Choquard in 1976, to study an electron trapped in its hole. Equation (1.2) can be used to describe many physical models. For instance, the quantum theory of a polaron [1], the modeling of an electron, a certain approximation to Hartree-Fock theory of one-component plasma [2], and the self-gravitational collapse of a quantum mechanical wave-function, etc. Moreover, the existence and qualitative properties of solutions to equation (1.2) have been widely studied in the last few decades, see e.g. [3,4,5, 6,7,8] for the work of Choquard-type equations.

Once we turn our attention to the Kirchhoff-type problems with critical nonlinearity, we immediately see that the literature is relatively scarce. In this case, we can cite the recent works of [9,10,11, 12,13,14, 15,16,17]. We call attention to [18] in which work the authors have dealt with the following Kirchhoff-type equation

(1.3) − a + b ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u + V λ ( x ) u = ( K μ ∗ u q ) ∣ u ∣ q − 2 u in R 3 .

By using the Nehari manifold and the concentration-compactness principle, the author obtained the existence of ground state solutions for equation (1.3) if the parameter λ is large enough.

On the other hand, another important reference is [19], where the authors have considered the following quasilinear Choquard equation:

− Δ u + V ( x ) u − [ Δ ( u 2 ) ] u = ( ∣ x ∣ − μ ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u in R N ,

where N ≥ 3 , μ ∈ ( 0 , ( N + 2 ) / 2 ) , p ∈ ( 2 , ( 4 N − 4 μ ) / ( N − 2 ) ) . By a changing variable and perturbation method, the existence of positive solutions, negative solutions, and high energy solutions was obtained. Moreover, we also cite previous studies [20,21,22, 23,24] with no attempt to provide the full list of references.

From the above mentioned papers, it is natural to ask what results can be recovered with this kind of quasilinear Choquard equation with critical nonlinearity in R N . Compared to the above papers, some difficulties arise in our paper when dealing with problem (1.1), because of the appearance of the convolution terms as well as critical nonlinearities which provokes some mathematical difficulties, and these make the study of problem (1.1) particularly interesting.

Our main results are as follows.

Theorem 1.1

Let 0 < μ < 4 and 1 < p < 4 . Suppose that Ω ≔ { x ∈ R N : k ( x ) > 0 } is an open subset of R N and that 0 < ∣ Ω ∣ < ∞ . Then, for each β > 0 there exists Λ ¯ > 0 such that if α ∈ ( 0 , Λ ¯ ) or for each α > 0 there exists Λ ̲ > 0 such that if β ∈ ( 0 , Λ ̲ ) , problem (1.1) has a sequence of solutions ( u n ) n and u n → 0 as n → ∞ in D 1 , 2 ( R N ) .

Theorem 1.2

Let 0 < μ < 4 , p = 4 , and β = 1 . Then, there exists a positive constant a ∗ such that, for each a > a ∗ and α ∈ 0 , 1 2 a S ‖ k ‖ r − 1 , problem (1.1) has at least k pairs of nontrivial weak solutions.

Remark 1.1

The difficulties of this paper mainly lie in two aspects: one of difficulties of the problem (1.1) stems from that there is no suitable working space on which the energy functional enjoys both smoothness and compactness, so the standard critical point theory cannot be applied directly. In order to overcome this difficulty, we use the method in [25, 26,27]. The other is caused by the convolution terms as well as critical nonlinearities, which leads to some estimates about nonlocal term that are likely to be confronted some difficulties. In order to prove the compactness condition, we use the second concentration-compactness principle and concentration-compactness principle at infinity to prove that the ( P S ) c condition holds.

2 Preliminaries

In this section, we first recall the following well-known Hardy-Littlewood-Sobolev inequality, which will be used in the sequel.

Proposition 2.1

[28] Let t , r > 1 and 0 < μ < N with 1 / t + μ / N + 1 / r = 2 , f ∈ L t ( R N ) , and h ∈ L r ( R N ) . There exists a sharp constant C ( t , r , μ , N ) independent of f , h , such that

(2.1) ∫ R N ∫ R N f ( x ) h ( y ) ∣ x − y ∣ μ d x d y ≤ C ( t , r , μ , N ) ‖ f ‖ L t ‖ h ‖ L r .

Let S be the best constant for the embedding D 1 , 2 ( R N ) into L 2 ∗ ( R N ) , that is,

(2.2) S = inf u ∈ D 1 , 2 ( R N ) ⧹ { 0 } ∫ R N ∣ ∇ u ∣ 2 d x : ∫ R N ∣ u ∣ 2 ∗ d x = 1 .

Consequently, we define

(2.3) S H , L = inf u ∈ D 1 , 2 ( R N ) ⧹ { 0 } ∫ R N ∣ ∇ u ∣ 2 d x : ∫ R N ∫ R N ∣ u ( x ) ∣ 2 μ ∗ ∣ u ( y ) ∣ 2 μ ∗ ∣ x − y ∣ μ d x d y = 1 .

Remark 2.1

From [4], we know that the constant S H , L defined in (2.3) is achieved, and

‖ ⋅ ‖ N L ≔ ∫ R N ∫ R N ∣ ⋅ ∣ 2 μ ∗ ∣ ⋅ ∣ 2 μ ∗ ∣ x − y ∣ μ d x d y 1 2 2 μ ∗

defines a norm on L 2 ∗ ( R N ) .

Problem (1.1) corresponding to the energy functional J : D 1 , 2 ( R N ) → R is defined by

J ( u ) ≔ a 2 ∫ R N ∣ ∇ u ∣ 2 d x + b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 + a ∫ R N ∣ u ∣ 2 ∣ ∇ u ∣ 2 d x − α p ∫ R N k ( x ) ∣ u ∣ p d x − β 2 2 μ ∗ ∫ R N ∫ R N ∣ u ( x ) ∣ 2 2 μ ∗ ∣ u ( y ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d x d y = a 2 ∫ R N ( 1 + 2 ∣ u ∣ 2 ) ∣ ∇ u ∣ 2 d x + b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 − α p ∫ R N k ( x ) ∣ u ∣ p d x − β 2 2 μ ∗ ∫ R N ∫ R N ∣ u ( x ) ∣ 2 2 μ ∗ ∣ u ( y ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d x d y .

Note that the functional J is not well defined in D 1 , 2 ( R N ) . In order to overcome this difficulty, we make the changing of variables v = f − 1 ( u ) , where f is defined by

f ′ ( t ) = 1 1 + 2 f 2 ( t ) , and f ( 0 ) = 0

on [ 0 , + ∞ ) and by f ( t ) = − f ( − t ) on ( − ∞ , 0 ] .

We have collected some properties of the function f .

Lemma 2.1

[25,29] The function f satisfies the following properties:

f is uniquely defined C ∞ and invertible.
∣ f ′ ( t ) ∣ ≤ 1 for all t ∈ R .
f ( t ) t → 1 as t → 0 .
f ( t ) t → 2 1 4 as t → ∞ .
1 2 f ( t ) ≤ t f ′ ( t ) ≤ f ( t ) for all t ≥ 0 .
1 2 f 2 ( t ) ≤ f ( t ) f ′ ( t ) t ≤ f 2 ( t ) for all t ∈ R .
∣ f ( t ) ∣ ≤ t for all t ∈ R .
∣ f ( t ) ∣ ≤ 2 1 4 ∣ t ∣ 1 2 for all t ∈ R .
The function f 2 ( t ) is strictly convex.
There exists a positive constant C such that
∣ f ( t ) ∣ ≥ C ∣ t ∣ , ∣ t ∣ ≤ 1 , C ∣ t ∣ 1 2 , ∣ t ∣ ≥ 1 .
There exist positive constants C 1 and C 2 such that
∣ t ∣ ≤ C 1 ∣ f ( t ) ∣ + C 2 ∣ f ( t ) ∣ 2 for all t ∈ R .
∣ f ( t ) f ′ ( t ) ∣ ≤ 1 2 for all t ∈ R .

So after the change of variables, we can write J ( u ) as

(2.4) J ( v ) ≔ a 2 ∫ R N ∣ ∇ v ∣ 2 d x + b 4 ∫ R N ∣ f ′ ( v ) ∣ 2 ∣ ∇ v ∣ 2 d x 2 − α p ∫ R N k ( x ) ∣ f ( v ) ∣ p d x − β 2 2 μ ∗ ∬ R 2 N ∣ f ( v ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d x d y .

By Proposition 2.1 and Lemma 2.1, we know that the functional J ∈ C 1 ( D 1 , 2 ( R N ) , R ) . As in [25], we note that if f is a nontrivial critical point of J , then v is a nontrivial solution of problem

(2.5) − a Δ v − b ∫ R N ∣ f ′ ( v ) ∣ 2 ∣ ∇ v ∣ 2 d x ⋅ ( f ′ ( v ) f ″ ( v ) ∣ ∇ v ∣ 2 + ∣ f ′ ( v ) ∣ 2 Δ v ) = g ( x , v ) ,

where

g ( x , s ) = f ′ ( s ) α k ( x ) ∣ f ( s ) ∣ p − 2 f ( s ) + β ∫ R N ∣ f ( v ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d y ∣ f ( v ) ∣ 2 2 μ ∗ − 2 f ( s ) .

Therefore, let u = f ( v ) and since ( f − 1 ) ′ ( t ) = [ f ′ ( f − 1 ( t ) ) ] − 1 = 1 + 2 t 2 , we conclude that u is a nontrivial solution of the problem

− a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u − a [ Δ ( u 2 ) ] u = α k ( x ) ∣ u ∣ p − 2 u + β ∫ R N ∣ u ∣ 2 2 μ ∗ ∣ x − y ∣ μ d y ∣ u ∣ 2 2 μ ∗ − 2 u .

3 The Palais-Smale condition

In this section, we will use the concentration-compactness principle for studying the critical Choquard equation [30] which is due to Lions [31] to prove the ( P S ) c condition.

Lemma 3.1

Let 1 < p < 4 . Then any ( P S ) c sequence { v n } is bounded in D 1 , 2 ( R N ) .

Proof

Let { v n } be a ( P S ) c sequence in D 1 , 2 ( R N ) such that

(3.1) c + o ( 1 ) = J ( v n ) = a 2 ∫ R N ∣ ∇ v n ∣ 2 d x + b 4 ∫ R N ∣ f ′ ( v n ) ∣ 2 ∣ ∇ v n ∣ 2 d x 2 − α p ∫ R N k ( x ) ∣ f ( v n ) ∣ p d x − β 2 2 μ ∗ ∬ R 2 N ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d x d y ,

(3.2) o ( 1 ) ‖ w ‖ = ⟨ J ′ ( v n ) , w ⟩ = a ∫ R N ∇ v n ∇ w d x − α ∫ R N k ( x ) ∣ f ( v n ) ∣ p − 2 f ( v n ) f ′ ( v n ) w d x + b ∫ R N ∣ ∇ v n ∣ 2 1 + 2 f 2 ( v n ) d x ∫ R N ∇ v n ∇ w ( 1 + 2 f 2 ( v n ) ) − 2 ∣ ∇ v n ∣ 2 f ( v n ) f ′ ( v n ) w [ 1 + 2 f 2 ( v n ) ] 2 d x − β ∬ R 2 N ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ − 2 f ( v n ( y ) ) f ′ ( v n ( y ) ) w ( y ) ∣ x − y ∣ μ d x d y .

Choose w = w n = 1 + 2 f 2 ( v n ) f ( v n ) , we have w n ∈ D 1 , 2 ( R N ) . From ( f 4 ) and since

∣ ∇ w n ∣ = 1 + 2 f 2 ( v n ) 1 + 2 f 2 ( v n ) ∣ ∇ v n ∣ ,

we deduce that ‖ w n ‖ ≤ c ‖ v n ‖ . By (3.2) we have

(3.3) o ( 1 ) ‖ v n ‖ = ⟨ J ′ ( v n ) , w n ⟩ = a ∫ R N 1 + 2 f 2 ( v n ) 1 + 2 f 2 ( v n ) ∣ ∇ v n ∣ 2 d x + b ∫ R N ∣ ∇ v n ∣ 2 1 + 2 f 2 ( v n ) d x 2 − α ∫ R N k ( x ) ∣ f ( v n ) ∣ p d x − β ∬ R 2 N ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d x d y .

Thus, using Hölder’s inequality and Sobolev embedding, and together with (3.1), (3.2), and (3.3), we have

c + o ( 1 ) ‖ v n ‖ = J ( v n ) − 1 2 2 μ ∗ ⟨ J ′ ( v n ) , w n ⟩ = a ∫ R N 1 2 − 1 2 2 μ ∗ 1 + 2 f 2 ( v n ) 1 + 2 f 2 ( v n ) ∣ ∇ v n ∣ 2 d x + 1 4 − 1 2 2 μ ∗ b ∫ R N ∣ f ′ ( v n ) ∣ 2 ∣ ∇ v n ∣ 2 d x 2 + 1 2 2 μ ∗ − 1 p α ∫ R N k ( x ) ∣ f ( v n ) ∣ p d x ≥ 1 2 − 1 2 μ ∗ a ∫ R N ∣ ∇ v n ∣ 2 d x − 1 p − 1 2 2 μ ∗ α ∫ R N ∣ k ( x ) ∣ r d x 1 r ∫ R N ∣ f ( v n ) ∣ 2 2 ∗ d x p 2 2 ∗ ≥ 1 2 − 1 2 μ ∗ a ∫ R N ∣ ∇ v n ∣ 2 d x − 1 p − 1 2 2 μ ∗ α ∫ R N ∣ k ( x ) ∣ r d x 1 r ∫ R N ∣ ∇ f 2 ( v n ) ∣ 2 d x p 4 ≥ 1 2 − 1 2 μ ∗ a ‖ v n ‖ 2 − c ‖ v n ‖ p 2 ,

which implies that { v n } is bounded in D 1 , 2 ( R N ) since 2 < 2 μ ∗ and 1 < p < 4 .□

Lemma 3.2

Let c < 0 , 0 < μ < 4 , and 1 < p < 4 . The next two properties hold.

For each β > 0 there exists Λ ¯ > 0 such that J satisfies the ( P S ) c condition for all α ∈ ( 0 , Λ ¯ ) .
For each α > 0 there exists Λ ̲ > 0 such that J satisfies the ( P S ) c condition for any β ∈ ( 0 , Λ ̲ ) .

Proof

Let { v n } ⊂ D 1 , 2 ( R N ) be a ( P S ) c -sequence. By Lemma 3.1, { v n } is bounded in D 1 , 2 ( R N ) . Then { f ( v n ) } is also bounded in D 1 , 2 ( R N ) . Therefore, we can assume that v n ⇀ v in D 1 , 2 ( R N ) , v n → v a.e. in R N , since f ∈ C ∞ , then f 2 ( v n ) → f 2 ( v ) a.e. in R N and then f 2 ( v n ) ⇀ f 2 ( v ) in D 1 , 2 ( R N ) . Hence, we can assume that

∣ ∇ f 2 ( v n ) ∣ 2 ⇀ ω , ∣ f ( v n ) ∣ 2 2 ∗ → ζ , ∫ R N ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d y ∣ f ( v n ) ∣ 2 2 μ ∗ ⇀ ν ,

where ω , ζ , and ν are bounded nonnegative measures on R N . By the concentration-compactness principle in [30], there exist at most countable sets I , sequences of points { x i } i ∈ I ⊂ R N , and families of positive numbers { ν i : i ∈ I } , { ω i : i ∈ I } , and { ζ i : i ∈ I } such that

(3.4) ν = ∫ R N ∣ f ( v ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d y ∣ f ( v ( y ) ) ∣ 2 2 μ ∗ + ∑ i ∈ I ν i δ x i ,

(3.5) ω ≥ ∣ ∇ f 2 ( v ) ∣ 2 + ∑ i ∈ I ω i δ x i , ζ ≥ ∣ f ( v ( y ) ) ∣ 2 2 ∗ + ∑ i ∈ I ζ i δ x i ,

(3.6) S H , L v i 1 2 μ ∗ ≤ ω i and ν i ≤ C ( N , μ ) ζ i 2 N − μ N ,

where δ x i is the Dirac mass at x i . Now, we take a smooth cut-off function φ ε , i centered at x i such that

0 ≤ φ ε , i ( x ) ≤ 1 , φ ε , i ( x ) = 1 in B x i , ε 2 , φ ε , i ( x ) = 0 in R N ⧹ B ( x i , ε ) , ∣ ∇ φ ε , i ( x ) ∣ ≤ 4 ε ,

for any ε > 0 small. Let w n = 1 + 2 f 2 ( v n ) f ( v n ) , then { w n } is bounded in D 1 , 2 ( R N ) and ⟨ J ′ ( v n ) , w n φ ε , i ⟩ → 0 . Thus,

(3.7) − a ∫ R N 1 + 2 f 2 ( v n ) f ( v n ) ∇ v n ∇ φ ε , i d x − b ∫ R N ∣ ∇ v n ∣ 2 1 + 2 f 2 ( v n ) d x ∫ R N f ( v n ) ∇ v n ∇ φ ε , i 1 + 2 f 2 ( v n ) d x = a ∫ R N 1 + 2 f 2 ( v n ) 1 + 2 f 2 ( v n ) ∣ ∇ v n ∣ 2 φ ε , i d x + b ∫ R N ∣ ∇ v n ∣ 2 1 + 2 f 2 ( v n ) d x ∫ R N ∣ ∇ v n ∣ 2 φ ε , i 1 + 2 f 2 ( v n ) d x − α ∫ R N k ( x ) ∣ f ( v n ) ∣ p φ ε , i d x − β ∬ R 2 N ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ φ ε , i ( y ) ∣ x − y ∣ μ d x d y + o n ( 1 ) .

The Hölder inequality and ( f 4 ) imply that

(3.8) 0 ≤ lim ε → 0 lim n → ∞ a ∫ R N 1 + 2 f 2 ( v n ) f ( v n ) ∇ v n ∇ φ ε , i d x ≤ c lim ε → 0 lim n → ∞ ∫ R N v n ∇ v n ∇ φ ε , i d x ≤ C lim ε → 0 lim n → ∞ ∫ R N ∣ ∇ v n ∣ 2 d x 1 2 ∫ R N ∣ v n ∇ φ ε , i ∣ 2 d x 1 2 ≤ C lim ε → 0 ∫ B ( x i , 2 ε ) ∣ v ∣ 2 ∗ d x 1 2 ∗ = 0 .

Similarly, we have

(3.9) lim ε → 0 lim n → ∞ b ∫ R N ∣ ∇ v n ∣ 2 1 + 2 f 2 ( v n ) d x ∫ R N f ( v n ) ∇ v n ∇ φ ε , i 1 + 2 f 2 ( v n ) d x = 0 .

From the definition of φ ε , i that

lim ε → 0 lim n → ∞ ∫ R N k ( x ) ∣ f ( v n ) ∣ p φ ε , i d x = 0 .

By (3.7)–(3.9), we get

(3.10) 0 = lim ε → 0 lim n → ∞ a ∫ R N 1 + 2 f 2 ( v n ) 1 + 2 f 2 ( v n ) ∣ ∇ v n ∣ 2 φ ε , i d x + b ∫ R N ∣ ∇ v n ∣ 2 1 + 2 f 2 ( v n ) d x ∫ R N ∣ ∇ v n ∣ 2 φ ε , i 1 + 2 f 2 ( v n ) d x − α ∫ R N k ( x ) ∣ f ( v n ) ∣ p φ ε , i d x − β ∬ R 2 N ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ φ ε , i ( y ) ∣ x − y ∣ μ d x d y ≥ lim ε → 0 lim n → ∞ a 2 ∫ R N φ ε , i ∣ ∇ f 2 ( v n ) ∣ 2 d x − β ∬ R 2 N ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ φ ε , i ( y ) ∣ x − y ∣ μ d x d y = a 2 ω i − β ν i .

This fact implies that a ω i ≤ 2 β ν i . Together with (3.6), we obtain

(3.11) ω i ≥ ( 2 − 1 β − 1 a S H , L 2 μ ∗ ) 1 2 μ ∗ − 1 or ω i = 0 .

If w i 0 ≥ ( 2 − 1 β − 1 a S H , L 2 μ ∗ ) 1 2 μ ∗ − 1 for i 0 ∈ I . From the Hölder inequality, the Sobolev embedding, and the Young inequality, we have

α ∫ R N k ( x ) ∣ f ( v ) ∣ p d x ≤ α ‖ k ‖ r S − p 2 ‖ f ( v ) ‖ p = 1 2 − 1 2 μ ∗ a 2 1 p − 1 2 2 μ ∗ − 1 p 2 ‖ f ( v ) ‖ p 1 2 − 1 2 μ ∗ a 2 1 p − 1 2 2 μ ∗ − 1 − p 2 α ‖ k ‖ r S − p 2 ≤ 1 2 − 1 2 μ ∗ a 2 1 p − 1 2 2 μ ∗ − 1 ‖ f ( v ) ‖ 2 + 2 − p 2 1 2 − 1 2 μ ∗ − 1 2 a S 1 p − 1 2 2 μ ∗ p 2 − p ‖ k ‖ r 2 2 − p α 2 2 − p . (3.12)

According to this fact, we have

(3.13) 0 > c = lim n → + ∞ J ( v n ) − 1 2 2 μ ∗ ⟨ J ′ ( v n ) , 1 + 2 f 2 ( v n ) f ( v n ) ⟩ = lim n → + ∞ a ∫ R N 1 2 − 1 2 2 μ ∗ 1 + 2 f 2 ( v n ) 1 + 2 f 2 ( v n ) ∣ ∇ v n ∣ 2 d x + 1 4 − 1 2 2 μ ∗ b ∫ R N ∣ f ′ ( v n ) ∣ 2 ∣ ∇ v n ∣ 2 d x 2 + 1 2 2 μ ∗ − 1 p α ∫ R N k ( x ) ∣ f ( v n ) ∣ p d x

≥ lim n → + ∞ 1 2 − 1 2 μ ∗ a ∫ R N ∣ ∇ v n ∣ 2 d x − 1 p − 1 2 2 μ ∗ α ∫ R N k ( x ) ∣ f ( v n ) ∣ p d x ≥ lim n → + ∞ 1 2 − 1 2 μ ∗ a 2 ∫ R N ∣ ∇ f 2 ( v n ) ∣ 2 d x − 1 p − 1 2 2 μ ∗ α ∫ R N k ( x ) ∣ f ( v n ) ∣ p d x ≥ 1 2 − 1 2 μ ∗ a 2 ‖ f ( v ) ‖ 2 + ∑ i ∈ I w i − 1 p − 1 2 2 μ ∗ α ∫ Ω k ( x ) ∣ f ( v ) ∣ p d x ≥ 1 2 − 1 2 μ ∗ a 2 w i 0 − 2 − p 2 1 2 − 1 2 μ ∗ − 1 2 a S p 2 − p 1 p − 1 2 2 μ ∗ 2 2 − p ‖ k ‖ r 2 2 − p α 2 2 − p ≥ 1 2 − 1 2 μ ∗ ( 2 − 1 a S H , L ) 2 μ ∗ 2 μ ∗ − 1 β − 1 2 μ ∗ − 1 − 2 − p 2 1 2 − 1 2 μ ∗ − 1 2 a S p 2 − p 1 p − 1 2 2 μ ∗ 2 2 − p ‖ k ‖ r 2 2 − p α 2 2 − p .

Thus, for any β > 0 , we can choose α 1 > 0 so small such that for every 0 < α < α 1 , the last term on the right-hand side above is greater than zero, which is a contradiction.

Similarly, if α > 0 is given, we take β 1 > 0 so small that for every β ∈ ( 0 , β 1 ) again the right-hand side of (3.13) is greater than zero. This gives the required contradiction. Consequently, ω i = 0 for all i ∈ I in (3.11).

To obtain the possible concentration of mass at infinity, similarly, we define a cut-off function ψ R in C ∞ ( R N ) such that ψ R = 0 in B R ( 0 ) , ψ R = 1 in R N ⧹ B R + 1 ( 0 ) , and ∣ ∇ ψ R ∣ ≤ 2 / R in R N . Let

ω ∞ = lim R → ∞ lim sup n → ∞ ∫ { x ∈ R N : ∣ x ∣ > R } ∣ ∇ u n ∣ 2 d x , ζ ∞ = lim R → ∞ lim sup n → ∞ ∫ { x ∈ R N : ∣ x ∣ > R } ∣ u n ∣ 2 ∗ d x ,

and

ν ∞ = lim R → ∞ lim sup n → ∞ ∫ { x ∈ R N : ∣ x ∣ > R } ( K μ ∗ ∣ u n ∣ 2 μ ∗ ) ∣ u n ∣ 2 μ ∗ d x .

Thus, the Hardy-Littlewood-Sobolev and the Hölder inequalities give

ν ∞ = lim R → ∞ lim n → ∞ ∫ R N ∫ R N ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d y ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ψ R ( y ) d x ≤ C ( N , μ ) lim R → ∞ lim n → ∞ ∣ f ( v n ) ∣ 2 ∗ 2 μ ∗ ∫ R N ∣ f ( v n ( x ) ) ∣ 2 ∗ ψ R ( y ) d x 2 μ ∗ 2 ∗ ≤ C ˆ ζ ∞ 2 μ ∗ 2 ∗ .

Therefore,

(3.14) 0 = lim R → ∞ lim n → ∞ ⟨ J ′ ( v n ) , ψ R w n ⟩ = lim R → ∞ lim n → ∞ a ∫ R N 1 + 2 f 2 ( v n ) 1 + 2 f 2 ( v n ) ∣ ∇ v n ∣ 2 ψ R d x + b ∫ R N ∣ ∇ v n ∣ 2 1 + 2 f 2 ( v n ) d x ∫ R N ∣ ∇ v n ∣ 2 ψ R 1 + 2 f 2 ( v n ) d x − α ∫ R N k ( x ) ∣ f ( v n ) ∣ p ψ R d x − β ∬ R 2 N ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ ψ R ( y ) ∣ x − y ∣ μ d x d y

≥ lim R → ∞ lim n → ∞ a 2 ∫ R N ψ R ∣ ∇ f 2 ( v n ) ∣ 2 d x − α ∫ R N k ( x ) ∣ f ( v n ) ∣ p ψ R d x − β ∬ R 2 N ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ ψ R ( y ) ∣ x − y ∣ μ d x d y ≥ a 2 ω ∞ − β C ˆ ζ ∞ 2 μ ∗ 2 ∗ .

Therefore, a 2 ω ∞ ≤ β C ˆ ζ ∞ 2 μ ∗ 2 ∗ . As the discussion in [30], we obtain

(3.15) ω ∞ ≥ 2 − 1 a S 2 μ ∗ 2 C ˆ − 1 β − 1 2 2 μ ∗ − 2 or ω ∞ = 0 .

As in (3.8) and (3.9), we have

(3.16) 0 > c ≥ 1 2 − 1 2 μ ∗ ( 2 − 1 a S ) 2 μ ∗ 2 μ ∗ − 2 C ˆ − 2 2 μ ∗ − 2 β − 2 2 μ ∗ − 2 − 2 − p 2 1 2 − 1 2 μ ∗ − 1 2 a S p 2 − p 1 p − 1 2 2 μ ∗ 2 2 − p ‖ k ‖ r 2 2 − p α 2 2 − p .

Thus, for any β > 0 , we choose α 2 > 0 so small that for every α ∈ ( 0 , α 2 ) the right-hand side of (3.16) is greater than zero, which is a contradiction.

Similarly, if α > 0 is given, we select β 2 > 0 so small that for every β ∈ ( 0 , β 2 ) the right-hand side of (3.12) is greater than zero. This gives the required contradiction. Therefore, ω ∞ = 0 in (3.11).

From the arguments above, put

Λ ¯ = min { α 1 , α 2 } and Λ ̲ = min { β 1 , β 2 } .

Then, for any c < 0 and β > 0 we have

ω i = 0 for all i ∈ I and ω ∞ = 0

for all α ∈ ( 0 , Λ ¯ ) .

Similarly, for any c < 0 and α > 0 we again have

ω i = 0 for all i ∈ I and ω ∞ = 0

for any β ∈ ( 0 , Λ ¯ ) .

Hence, as n → ∞

∬ R 2 N ∣ v n ( x ) ∣ 2 μ ∗ ∣ v n ( y ) ∣ 2 μ ∗ ∣ x − y ∣ μ d x d y → ∬ R 2 N ∣ v ( x ) ∣ 2 μ ∗ ∣ v ( y ) ∣ 2 μ ∗ ∣ x − y ∣ μ d x d y

and

∫ R N k ( x ) ( ∣ v n ∣ q − ∣ v ∣ q ) d x ≤ ‖ k ‖ r ‖ ∣ v n ∣ q − ∣ v ∣ q ‖ 2 μ ∗ q → 0 .

Since ( ‖ v n ‖ ) n is bounded and J ′ ( v ) = 0 , the weak lower semicontinuity of the norm and the Brézis-Lieb lemma yield as n → ∞

o ( 1 ) ‖ v n ‖ = ⟨ J ′ ( v n ) , w n ⟩ = a ∫ R N 1 + 2 f 2 ( v n ) 1 + 2 f 2 ( v n ) ∣ ∇ v n ∣ 2 d x + b ∫ R N ∣ ∇ v n ∣ 2 1 + 2 f 2 ( v n ) d x 2 − α ∫ R N k ( x ) ∣ f ( v n ) ∣ p d x − β ∫ R N ∫ R N ∣ f ( v n ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v n ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d x d y = a ‖ v n ‖ + a ∫ R N 2 f 2 ( v n ) 1 + 2 f 2 ( v n ) ∣ ∇ v n ∣ 2 d x + b ∫ R N ∣ ∇ v n ∣ 2 1 + 2 f 2 ( v n ) d x 2

− α ∫ R N k ( x ) ∣ f ( v n ) ∣ p d x − β ∫ R N ∫ R N ∣ f ( v ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d x d y ≥ a ‖ v n − v ‖ 2 + a ‖ v ‖ 2 + a ∫ R N 2 f 2 ( v ) 1 + 2 f 2 ( v ) ∣ ∇ v ∣ 2 d x + b ∫ R N ∣ ∇ v ∣ 2 1 + 2 f 2 ( v ) d x 2 − α ∫ R N k ( x ) ∣ f ( v ) ∣ p d x − β ∫ R N ∫ R N ∣ f ( v ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d x d y = ‖ v n − v ‖ 2 + o ( 1 ) ‖ v ‖ .

This fact implies that { v n } strongly converges to v in D 1 , 2 ( R N ) . This completes the proof of Lemma 3.2.□

4 Proof of Theorems 1.1

In this section, we will use the following version of the symmetric mountain-pass lemma to prove the existence of infinitely many solutions of (1.1) which tend to zero.

Lemma 4.1

[32] Let E be an infinite-dimensional Banach space and J ∈ C 1 ( E , R ) . Suppose that the following properties hold.

J is even, bounded from below in E , J ( 0 ) = 0 and J satisfies the local Palais-Smale condition.
For each n ∈ N there exists A n ∈ Σ n such that sup u ∈ A n J ( u ) < 0 , where
Σ n ≔ { A : A ⊂ E is closed symmetric , 0 ∉ A , γ ( A ) ≥ n }
and γ ( A ) is a genus of A .

Then J admits a sequence of critical points ( u n ) n such that J ( u n ) ≤ 0 , u n ≠ 0 for each n and ( u n ) n converges to zero as n → ∞ .

Note that

J ( v ) ≥ a 2 ‖ v ‖ 2 − α p ∫ R N ∣ k ( x ) ∣ r d x 1 r ∫ R N f 2 ( 2 ∗ ) ( v ) d x p 2 ( 2 ∗ ) − S H , L − 1 2 2 μ ∗ β ‖ f ( v ) ‖ 2 2 μ ∗ ≥ a 2 ‖ v ‖ 2 − α c 1 ‖ f ( v ) ‖ p 2 − β c 2 ‖ f ( v ) ‖ 2 2 μ ∗ ≥ a 2 ‖ v ‖ 2 − α c 1 ‖ v ‖ p 2 − β c 2 ‖ v ‖ 2 2 μ ∗ ≕ Q ( ‖ v ‖ ) ,

where c 1 and c 2 are some positive constants and Q ( t ) ≔ a 2 t 2 − α c 1 t p 2 − β c 2 t 2 2 ∗ . Obviously, fixed β > 0 there exists α 1 > 0 so small that for every 0 < α < α 1 , there exists 0 < t 0 < t 1 , Q ( t ) > 0 for t 0 < t < t 1 , Q ( t ) < 0 for t > t 1 and 0 < t < t 0 . Similarly, fixed α > 0 , we can choose β 1 > 0 with the property that t 0 , t 1 as above exist for each 0 < β < β 1 . Clearly, Q ( t 0 ) = 0 = Q ( t 1 ) . Following the same idea as in [34], we consider the truncated functional

(4.1) J ˜ ( v ) = a 2 ∫ R N ∣ ∇ v ∣ 2 d x + b 4 ∫ R N ∣ f ′ ( v ) ∣ 2 ∣ ∇ v ∣ 2 d x 2 − α p ∫ R N k ( x ) ∣ f ( v ) ∣ p d x − β 2 2 μ ∗ φ ( v ) ∬ R 2 N ∣ f ( v ( x ) ) ∣ 2 2 μ ∗ ∣ f ( v ( y ) ) ∣ 2 2 μ ∗ ∣ x − y ∣ μ d x d y ,

where φ ( v ) = χ ( ‖ v ‖ ) and χ : R + → [ 0 , 1 ] is a nonincreasing C ∞ function such that χ ( t ) = 1 if t ≤ T 0 and χ ( t ) = 0 if t ≥ T 1 . Thus,

J ˜ ( v ) ≥ Q ¯ ( ‖ v ‖ ) ,

where Q ¯ ( t ) = a 2 t 2 − α c 1 t p 2 − β c 2 t 2 2 ∗ φ ( t ) . It is clear that J ˜ ( v ) ∈ C 1 and is bounded from below in D 1 , 2 ( R N ) .

From the above arguments, we have the next results for functional J ˜ ( v ) .

Lemma 4.2

Let J ˜ ( v ) be defined as in (4.1). Then

If J ˜ ( v ) < 0 , then ‖ v ‖ ≤ T 0 and J ˜ ( v ) = J ( v ) .
Let c < 0 . Then, for any β > 0 there exists Λ ¯ > 0 such that J ˜ satisfies the ( P S ) c condition for all α ∈ ( 0 , Λ ¯ ) .
Let c < 0 . Then, for any α > 0 there exists Λ ̲ > 0 such that J ˜ satisfies the ( P S ) c condition for all β ∈ ( 0 , Λ ̲ ) .

Proof of Theorem 1.1

Clearly, J ˜ ( 0 ) = 0 , and J ˜ is of class C 1 ( D 1 , 2 ( R N ) ) , even, coercive, and bounded from below in D 1 , 2 ( R N ) . Furthermore, J ˜ satisfies the ( P S ) c condition in D 1 , 2 ( R N ) , with c < 0 , by Lemma 4.2.

For any n ∈ N , we take n disjointing open sets X i such that ∪ i = 1 n X i ⊂ Ω , where Ω is the nonempty open set introduced in the statement of Theorem 1.1. For each i = 1 , 2 , ⋯ , n , take v i ∈ ( D 1 , 2 ( R N ) ∩ C 0 ∞ ( X i ) ) ⧹ { 0 } , with ‖ v i ‖ = 1 . Put E n = span { v 1 , v 2 , ⋯ , v n } .

Thus, for any v ∈ E n , with ‖ v ‖ = ρ , we have

J ˜ ( v ) ≤ a 2 ‖ v ‖ 2 + b 4 ‖ v ‖ 4 − α p ∫ Ω k ( x ) ∣ v ∣ p d x − β 2 ⋅ 2 μ ∗ ‖ v ‖ N L 2 ⋅ 2 μ ∗ ≤ a 2 ρ 2 + b 4 ρ 4 − C 1 ρ p − C 2 ρ 2 ⋅ 2 μ ∗ ,

where C 1 and C 2 are some positive constants, since all the norms are equivalent in the finite dimensional space E n . Hence, J ˜ ( v ) < 0 provided that ρ > 0 is sufficiently small, being 1 < q < 4 . Therefore,

{ u ∈ E n : ‖ v ‖ = ρ } ⊂ { v ∈ E n : J ˜ ( v ) < 0 } .

Moreover,

γ ( { v ∈ E n : ‖ v ‖ = ρ } ) = n .

Hence by the monotonicity of the genus γ , we have

γ ( { v ∈ E n : J ˜ ( v ) < 0 } ) ≥ n .

Choosing A n = { v ∈ E n : J ˜ ( v ) < 0 } , we have A n ∈ ∑ n and sup v ∈ A n J ˜ ( v ) < 0 . Therefore, all the assumptions of Lemma 4.1 are satisfied, since D 1 , 2 ( R N ) is a real infinite Hilbert space. Thus, there exists a sequence { v n } in D 1 , 2 ( R N ) such that

J ˜ ( v n ) ≤ 0 , v n ≠ 0 , J ˜ ′ ( v n ) = 0 for each n and ‖ v n ‖ → 0 as n → ∞ .

Combining with Lemma 4.2 and taking n so large that ‖ v n ‖ ≤ ρ is small enough, then these infinitely many nontrivial functions v n are solutions of (2.5). This completes the proof of Theorem 1.1, since u m = f ( v m ) ≠ u n = f ( v n ) if v m ≠ v n and f ∈ C ∞ .□

5 Proof of Theorem 1.2

In this section, we give the following general version of the mountain pass lemma in [33], which will be used to prove Theorem 1.2.

Proposition 5.1

[33] Let X be an infinite dimensional Banach space with X = V ⊕ Y , where V is finite dimensional and let J ∈ C 1 ( X , R ) be an even functional with J ( 0 ) = 0 such that the following conditions hold:

There exist positive constants ϱ , ρ > 0 such that J ( u ) ≥ ϱ for all u ∈ ∂ B ρ ( 0 ) ∩ Y .
There exists c ∗ > 0 such that J satisfies the ( P S ) c condition for 0 < c < c ∗ .
For each finite dimensional subspace X ^ ⊂ X , there exists R = R ( X ^ ) such that J ( u ) ≤ 0 for all u ∈ X ^ \ B R ( 0 ) .

Suppose that V is k dimensional and V = span { e 1 , e 2 , … , e k } . For n ≥ k , inductively choose e n + 1 ∉ X n ≔ span { e 1 , e 2 , … , e n } . Let R n = R ( X n ) and D n = B R n ( 0 ) ∩ X n . Define

G n ≔ { h ∈ C ( D n , X ) : h is odd and h ( u ) = u , ∀ ∂ B R n ( 0 ) ∩ X n }

and

Γ j ≔ { h ( D n \ E ¯ ) : h ∈ G n , n ≥ j , E ∈ ∑ and γ ( E ) ≤ n − j } .

For each j ∈ N , let

c j ≔ inf K ∈ Γ j max u ∈ K J ( u ) .

Then, 0 < ϱ ≤ c j ≤ c j + 1 for j > k , and if j > k and c j < c ∗ , then we conclude that c j is the critical value of J . Moreover, if c j = c j + 1 = … = c j + l = c < c ∗ for j > k , then γ ( K c ) ≥ l + 1 , where

K c ≔ { u ∈ E : J ( u ) = c and J ′ ( u ) = 0 } .

Lemma 5.1

Let α ∈ 0 , 1 2 a S 2 ‖ k ‖ L r − 1 . Then J satisfies ( P S ) c condition, for all c ∈ ( 0 , c ∗ ) , where

(5.1) c ∗ ≔ min 1 4 − 1 2 2 μ ∗ ( 2 − 1 a S H , L ) 2 μ ∗ 2 μ ∗ − 1 , 1 4 − 1 2 2 μ ∗ ( 2 − 1 a S ) 2 μ ∗ 2 μ ∗ − 2 C ˆ − 2 2 μ ∗ − 2 .

Proof

On the one hand, from Hölder’s inequality, Sobolev embedding theorem and ( f 7 ) , we get

(5.2) ∫ R N k ( x ) ∣ f ( v ) ∣ 4 d x ≤ 2 S − 2 ‖ k ‖ L r ‖ v ‖ 2 .

Together α ∈ 0 , 1 2 a S 2 ‖ k ‖ L r − 1 with (5.2), and proceeding as in proof of Lemma 3.1, we have

c + o ( 1 ) ‖ v n ‖ = J ( v n ) − 1 2 2 μ ∗ ⟨ J ′ ( v n ) , w n ⟩ ≥ 1 2 − 1 2 μ ∗ a ∫ R N ∣ ∇ v n ∣ 2 d x − 1 4 − 1 2 2 μ ∗ α ∫ R N k ( x ) ∣ f ( v n ) ∣ 4 d x ≥ 1 2 − 1 2 μ ∗ a ‖ v n ‖ 2 − 1 2 − 1 2 μ ∗ α S − 2 ‖ k ‖ L r ‖ v n ‖ 2 > 1 4 − 1 2 2 μ ∗ a ‖ v n ‖ 2 .

This fact implies that { v n } is bounded since 2 < 2 μ ∗ . As similar discussion in Lemma 3.2, we deduce that (3.11) and (3.15) hold. By contradiction, we assume that w i 0 ≥ ( 2 − 1 a S H , L 2 μ ∗ ) 1 2 μ ∗ − 1 for i 0 ∈ I and ω ∞ ≥ 2 − 1 a S 2 μ ∗ 2 C ˆ − 1 2 2 μ ∗ − 2 hold. Similar to Lemma 3.2, we deduce

(5.3) c = lim n → + ∞ J ( v n ) − 1 2 2 μ ∗ ⟨ J ′ ( v n ) , 1 + 2 f 2 ( v n ) f ( v n ) ⟩

≥ lim n → + ∞ 1 2 − 1 2 μ ∗ a ∫ R N ∣ ∇ v n ∣ 2 d x − 1 2 − 1 2 μ ∗ α S − 2 ‖ k ‖ L r ∫ R N ∣ ∇ v n ∣ 2 d x ≥ lim n → + ∞ 1 4 − 1 2 2 μ ∗ a 2 ∫ R N ∣ ∇ f 2 ( v n ) ∣ 2 d x ≥ 1 4 − 1 2 2 μ ∗ a 2 w i 0 ≥ 1 4 − 1 2 2 μ ∗ ( 2 − 1 a S H , L ) 2 μ ∗ 2 μ ∗ − 1

and

(5.4) c ≥ 1 4 − 1 2 2 μ ∗ ( 2 − 1 a S ) 2 μ ∗ 2 μ ∗ − 2 C ˆ − 2 2 μ ∗ − 2 .

Then, for any c ∈ ( 0 , c ∗ ) , (5.3) and (5.4) cannot happen. Thus, we have

ω i = 0 for all i ∈ I and ω ∞ = 0 .

The rest of the proof is the same as in the proof to Lemma 3.2. Therefore, the compactness of the Palais-Smale sequence holds.□

Remark 5.1

It is easy to verify that the functional J satisfies the hypotheses ( I 1 ) and ( I 3 ) for α ∈ 0 , 1 2 a S 2 ‖ k ‖ L r − 1 .

Lemma 5.2

There exists a sequence { M n } ⊂ ( 0 , + ∞ ) independent of α , with M n ≤ M n + 1 , such that for any α > 0 ,

c n α ≔ inf K ∈ Γ n max u ∈ K J ( v ) < M n .

Proof

Our proof is similar to that presented in [35, Lemma 5]. The definition of c n α implies that

c n α = inf K ∈ Γ n max v ∈ K J ( v ) ≤ inf K ∈ Γ n max v ∈ K a 2 ‖ v ‖ 2 + b 4 ‖ v ‖ 4 − 1 2 2 μ ∗ ∬ R 2 N ∣ f ( v ( x ) ) ∣ 2 μ ∗ ∣ f ( v ( y ) ) ∣ 2 μ ∗ ∣ x − y ∣ μ d x d y .

Let

M n ≔ inf K ∈ Γ n max v ∈ K a 2 ‖ v ‖ 2 + b 4 ‖ v ‖ 4 − 1 2 2 μ ∗ ∬ R 2 N ∣ f ( v ( x ) ) ∣ 2 μ ∗ ∣ f ( v ( y ) ) ∣ 2 μ ∗ ∣ x − y ∣ μ d x d y ,

then we conclude that M n < + ∞ and M n ≤ M n + 1 by the definition of Γ n .□

Proof of Theorem 1.2

Taking a ∗ > 0 large enough such that for any a > a ∗ , we have

0 < c 1 α ≤ c 2 α ≤ ⋯ ≤ c k α < M k < c ∗ .

From Proposition 5.1, the levels c 1 α ≤ c 2 α ≤ ⋯ ≤ c k α are critical values of J . So, if c 1 α < c 2 α < ⋯ < c k α , the functional J has at least k critical points. Now, if c j α = c j + 1 α for some j = 1 , 2 , … , k − 1 , again Proposition 5.1 implies that K c j α is an infinite set (see [33, Chapter 7]) and hence in this case, problem (5.1) has infinitely many weak solutions. Consequently, problem (5.1) has at least k pair of weak solutions. Therefore, problem (2.5) has at least k pairs of solutions and u = f ( v ) must solve problem (1.1).□

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions.

Funding information: R. Li was supported by the Science and Technology Development Plan Project of Jilin Province (No. 20200401123GX). Y. Song was supported by the National Natural Science Foundation of China (No. 12001061) and the Research Foundation of Department of Education of Jilin Province (JJKH20220827KJ).
Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflict of interest: Authors state no conflict of interest.

References

[1] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. 10.1515/9783112649305Search in Google Scholar

[2] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’ nonlinear equation, Stud. Appl. Math. 57 (1977), no. 2, 93–105, https://doi.org/10.1002/sapm197757293. Search in Google Scholar

[3] F. Gao and M. Yang, On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents, J. Math. Anal. Appl. 448 (2017), no. 2, 1006–1041, https://doi.org/10.1016/j.jmaa.2016.11.015. Search in Google Scholar

[4] F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math. 61 (2018), no. 7, 1219–1242, https://doi.org/10.1007/s11425-016-9067-5. Search in Google Scholar

[5] J. Giacomoni, T. Mukherjee, and K. Sreenadh, Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity, J. Math. Anal. Appl. 467 (2018), no. 1, 638–672, https://doi.org/10.1016/j.jmaa.2018.07.035. Search in Google Scholar

[6] V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579, https://doi.org/10.1090/S0002-9947-2014-06289-2. Search in Google Scholar

[7] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), no. 5, 1550005, https://doi.org/10.1142/S0219199715500054. Search in Google Scholar

[8] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), no. 2, 153–184, https://doi.org/10.1016/j.jfa.2013.04.007. Search in Google Scholar

[9] D. Cassani and J. Zhang, Choquard-type equations with Hardy-Littlewood-Sobolev upper-critical growth, Adv. Nonlinear Anal. 8 (2019), no. 1, 1184–1212, https://doi.org/10.1515/anona-2018-0019. Search in Google Scholar

[10] G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013), no. 2, 706–713, https://doi.org/10.1016/j.jmaa.2012.12.053. Search in Google Scholar

[11] G. M. Figueiredo and J. R. Junior, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth, Differ. Integral Equ. 25 (2012), no. 9–10, 853–868.10.57262/die/1356012371Search in Google Scholar

[12] S. Liang and S. Shi, Soliton solutions to Kirchhoff type problems involving the critical growth in RN, Nonlinear Anal. 81 (2013), 31–41, https://doi.org/10.1016/j.na.2012.12.003. Search in Google Scholar

[13] S. Liang and J. Zhang, Existence of solutions for Kirchhoff type problems with critical nonlinearity in R3, Nonlinear Anal. Real World Appl. 17 (2014), 126–136, https://doi.org/10.1016/j.nonrwa.2013.10.011. Search in Google Scholar

[14] S. Liang and J. Zhang, Multiplicity of solutions for the noncooperative Schrödinger-Kirchhofi system involving the fractional p-Laplacian in RN, Z. Angew. Math. Phys. 68 (2017), art. 63, https://doi.org/10.1007/s00033-017-0805-9. Search in Google Scholar

[15] S. Liang, P. Pucci, and B. Zhang, Multiple solutions for critical Choquard-Kirchhoff type equations, Adv. Nonlinear Anal. 10 (2021), no. 1, 400–419, https://doi.org/10.1515/anona-2020-0119. Search in Google Scholar

[16] S. Liang, D. Repovs, and B. Zhang, Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities, Math. Models Methods Appl. Sci. 43 (2020), no. 5, 2473–2490, https://doi.org/10.1002/mma.6057. Search in Google Scholar

[17] M. Xiang, B. Zhang, and V. Rădulescu, Superlinear Schrödinger-Kirchhoff type problems involving the fractional p-Laplacian and critical exponent, Adv. Nonlinear Anal. 9 (2020), no. 1, 690–709, https://doi.org/10.1515/anona-2020-0021. Search in Google Scholar

[18] D. F. Lü, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal. 99 (2014), no. 99, 35–48, https://doi.org/10.1016/j.na.2013.12.022. Search in Google Scholar

[19] X. Yang, W. Zhang, and F. Zhao, Existence and multiplicity of solutions for a quasilinear Choquard equation via perturbation method, J. Math. Phys. 59 (2018), 081503, https://doi.org/10.1063/1.5038762. Search in Google Scholar

[20] C. O. Alves and M. Yang, Multiplicity and concentration of solutions for a quasilinear Choquard equation, J. Math. Phys. 55 (2014), 061502, https://doi.org/10.1063/1.4884301. Search in Google Scholar

[21] S. Chen and X. Wu, Existence of positive solutions for a class of quasilinear Schrödinger equations of Choquard type, J. Math. Anal. Appl. 475 (2019), no. 2, 1754–1777, https://doi.org/10.1016/j.jmaa.2019.03.051. Search in Google Scholar

[22] J. Lee, J. M. Kim, J. H. Bae, and K. Park, Existence of nontrivial weak solutions for a quasilinear Choquard equation, J. Inequal. Appl. 2018 (2018), art. 42, https://doi.org/10.1186/s13660-018-1632-z. 10.1186/s13660-018-1632-zSearch in Google Scholar PubMed PubMed Central

[23] X. Yang, X. Tang, and G. Gu, Multiplicity and concentration behavior of positive solutions for a generalized quasilinear Choquard equation, Complex Var. Elliptic Equ. 65 (2020), no. 9, 1515–1547, https://doi.org/10.1080/17476933.2019.1664487. 10.1080/17476933.2019.1664487Search in Google Scholar

[24] W. Zhang and X. Wu, Existence, multiplicity, and concentration of positive solutions for a quasilinear Choquard equation with critical exponent, J. Math. Phys. 60 (2019), no. 5, 051501, https://doi.org/10.1063/1.5051205. Search in Google Scholar

[25] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. 56 (2004), no. 2, 213–226. 10.1016/j.na.2003.09.008Search in Google Scholar

[26] J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, Soliton solutions to quasilinear Schrödinger equations II, J. Differ. Equ. 187 (2003), no. 2, 473–493. 10.1016/S0022-0396(02)00064-5Search in Google Scholar

[27] J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Diff. Eqn. 29 (2004), no. 5–6, 879–901. 10.1081/PDE-120037335Search in Google Scholar

[28] E. Lieb and M. Loss, Analysis, 2nd edn., Graduate Studies in Mathematics, vol. 14, AMS, Providence, Rhode Island, 2001. 10.1090/gsm/014Search in Google Scholar

[29] J. Marcos do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal. 8 (2009), no. 2, 621–644, https://doi.org/10.3934/cpaa.2009.8.621. Search in Google Scholar

[30] F. Gao, E. D. da Silva, M. Yang, and J. Zhou, Existence of solutions for critical Choquard equations via the concentration compactness method, Proc. Roy. Soc. Edinb. A 150 (2020), no. 2, 921–954, https://doi.org/10.1017/prm.2018.131. Search in Google Scholar

[31] P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. H. Poincare Anal. Non. Lineaire 1 (1984), no. 2, 109–145 and 223–283, https://doi.org/10.1016/S0294-1449(16)30428-0. 10.1016/s0294-1449(16)30428-0Search in Google Scholar

[32] R. Kajikiya, A critical-point theorem related to the symmetric mountain-pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), no. 2, 352–370, https://doi.org/10.1016/j.jfa.2005.04.005. Search in Google Scholar

[33] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: CBMS Regional Conference Series in Mathematics, vol. 65, 1984. 10.1090/cbms/065Search in Google Scholar

[34] J. Garcia Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differ. Equ. 144 (1998), no. 2, 441–476, https://doi.org/10.1006/jdeq.1997.3375. Search in Google Scholar

[35] Z. Wei and X. Wu, A multiplicity result for quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 18 (1992), no. 6, 559–567, https://doi.org/10.1016/0362-546X(92)90210-6. 10.1016/0362-546X(92)90210-6Search in Google Scholar

Received: 2021-08-24

Accepted: 2021-10-27

Published Online: 2021-12-31

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

Articles in the same Issue

https://doi.org/10.1515/math-2021-0125

Keywords for this article

quasilinear Choquard equation; Hardy-Littlewood-Sobolev critical exponent; Mountain-pass theorem; concentration-compactness principle

Creative Commons

BY 4.0