Startseite Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
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Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents

  • Xiaolu Lin , Shenzhou Zheng und Zhaosheng Feng EMAIL logo
Veröffentlicht/Copyright: 29. April 2023
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Advanced Nonlinear Studies
Aus der Zeitschrift Advanced Nonlinear Studies Band 23 Heft 1

Abstract

In this article, we are concerned with multiple solutions of Schrödinger-Choquard-Kirchhoff equations involving the fractional p -Laplacian and Hardy-Littlewood-Sobolev critical exponents in R N . We classify the multiplicity of the solutions in accordance with the Kirchhoff term M ( ⋅ ) and different ranges of q shown in the nonlinearity f ( x , ⋅ ) by means of the variational methods and Krasnoselskii’s genus theory. As an immediate consequence, some recent related results have been improved and extended.

Keywords: fractional p-Laplacian; multiple solutions; Hardy-Littlewood-Sobolev’s critical exponents; concentration-compactness; genus theory
MSC 2010: 35A15; 35J60; 35J20

1 Introduction

Let p ∈ ( 1 , ∞ ) , μ ∈ ( 0 , N ) , and N > p s with s ∈ ( 0 , 1 ) . We consider the multiplicity of the solutions for a class of Schrödinger-Choquard-Kirchhoff-type equations with the fractional p -Laplacian in R N :

(1) M ( ‖ u ‖ V p ) [ ( − Δ ) p s u + V ( x ) ∣ u ∣ p − 2 u ] = α f ( x , u ) + β ∫ R N ∣ u ( y ) ∣ p s , μ ∗ ∣ x − y ∣ μ d y ∣ u ∣ p s , μ ∗ − 2 u ,

where α and β are positive real parameters, f represents the nonlinearity, p s , μ ∗ = N p − μ p / 2 N − s p is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, and

‖ u ‖ V = ∬ R 2 N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + ∫ R N V ( x ) ∣ u ∣ p d x 1 p .

The fractional p -Laplacian operator ( − Δ ) p s is defined (up to normalization factors) for x ∈ R N as follows:

( − Δ ) p s φ = 2 lim δ → 0 + ∫ R N ⧹ B δ ( x ) ∣ φ ( x ) − φ ( y ) ∣ p − 2 ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + s p d y , φ ∈ C 0 ∞ ( R N ) ,

where B δ ( x ) indicates an open ball of R N centered at x with the radius δ > 0 .

The study of the existence for Choquard type equations, in recent years, has attracted continuous attention from a rather diverse group of scientists, such as physicists and mathematicians, due to its widespread applications in scientific areas. As we know, the nonlinear Choquard or Choquard-Pekar equation that can be traced back to Pekar [35], describes the polaron at rest in quantum theory

(2) − Δ u + u = ( I α ∗ ∣ u ∣ 2 ) u , x ∈ R 3 ,

which is elaborated to characterize as a certain approximation of the Hartree-Fock theory in terms of plasma, where I α ( x ) = ∣ x ∣ − ( n − α ) for α > 0 is a kernel function. Penrose [36] later proposed the time-dependent form as a model of the self-gravitational collapse of a quantum mechanical wave function. Quite a few profound results on the qualitative properties of the solutions to Choquard-type equations have been presented. Especially, as an important pioneering work, Lieb [24] applied the variational methods to investigate the existence and uniqueness, up to translations, of positive solutions to equation (2) in R 3 . Lions [28,29] established the existence of a sequence of radially symmetric solutions to this equation. For more results on the existence, regularity, positivity, radially symmetric, and ground state solutions of Choquard-type equations, we refer to [16–18,25,32,33,41]. For the critical case of the Hardy-Littlewood-Sobolev inequality, we refer to [5,11,14,27,31] for recent results in a smooth bounded domain of R N . We would also like to mention that some new progress on weighted Hardy-Littlewood-Sobolev inequality and Stein-Weiss inequality can be referred to in the literatures [6–8,42].

The Kirchhoff-type equations arise from various physical and biological phenomena, which were extensively studied in the past decades [1,19,39,45]. For example, among them, we have seen the Kirchhoff-Choquard problem with critical growth [15], and the fractional Schrödinger-Kirchhoff equation [2], the fractional p -Kirchhoff equation with subcritical growth [4], the fractional p -Laplacian equation of Schrödinger-Choquard-Kirchhoff-type [38], the Choquard-Kirchhoff equation with Hardy-Littlewood-Sobolev critical exponent [21], the fractional p and q Laplacian problem with critical growth [3], the degenerate Kirchhoff fractional ( p , q ) system [12] and the fractional ( p , q ) -Kirchhoff equation with critical Sobolev-Hardy exponent [26] and with singular and exponential nonlinearities [34] etc.

The main purpose of this article is to study the multiplicity of the solutions for the fractional Schrödinger-Choquard-Kirchhoff equation with the Hardy-Littlewood-Sobolev critical nonlinearity in R N . It is worth emphasizing that this is not a trivial problem, since there is an upper critical exponent in the Hardy-Littlewood-Sobolev inequality, a nonlocal nature of the fractional p -Laplacian, and the presence of potential V ( x ) . To the best of our knowledge, very little has been undertaken on equation (1) in the literature.

Throughout this article, we assume that the potential V ( ⋅ ) and the Kirchhoff term M ( ⋅ ) are equipped with the following hypotheses:

( V 1 ) V ∈ C ( R N ) satisfies V ( x ) ≥ V 0 > 0 , where V 0 > 0 is a constant.

( V 2 ) There exists ς > 0 such that lim ∣ y ∣ → ∞ meas { x ∈ B ς ( y ) : V ( x ) ≤ c } = 0 for c > 0 .

( M 1 ) M ( t ) ∈ C ( R 0 + ) satisfies M ( t ) ≥ m 0 > 0 , where m 0 is a constant.

( M 2 ) There exists θ ∈ [ 1 , 2 p s , μ ∗ / p ) such that θ M ( t ) ≔ θ ∫ 0 t M ( τ ) d τ ≥ M ( t ) t for t ∈ R 0 + .

Definition 1

We say that u ∈ W V s , p ( R N ) is a weak solution of equation (1), if for all ϕ ∈ W V s , p ( R N ) , there holds

M ( ‖ u ‖ V p ) ⟨ u , ϕ ⟩ s , p + ∫ R N V ( x ) ∣ u ∣ p − 2 u ϕ d x = α ∫ R N f ( x , u ) ϕ d x + β ∫ R N ∫ R N ∣ u ( y ) ∣ p s , μ ∗ ∣ x − y ∣ μ d y ∣ u ∣ p s , μ ∗ − 2 u ϕ d x ,

where

⟨ u , ϕ ⟩ s , p = ∬ R 2 N ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ( ϕ ( x ) − ϕ ( y ) ) ∣ x − y ∣ N + p s d x d y .

The energy functional J α , β : W V s , p ( R N ) → R N associated with equation (1) is well-defined as follows:

J α , β ( u ) = 1 p M ( ‖ u ‖ V p ) − α ∫ R N F ( x , u ) d x − β 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ( x ) ∣ p s , μ ∗ d x ,

where F ( x , t ) is the antiderivative of f ( x , t ) shown as in ( F 2 ) below, and K μ ( x ) = ∣ x ∣ − μ . It is easy to check that J α , β ∈ C 1 ( R N , R ) , whose critical points are solutions of equation (1).

Before summarizing the main results on the multiplicity of the solutions, we assume that the nonlinearity f ( x , u ) satisfies the following condition:

( F 0 ) Suppose that f ( x , u ) = h ( x ) ∣ u ∣ q − 2 u , where h ( x ) is a nonnegative function satisfying 0 ≢ h ( x ) ∈ L r ( R N ) with r = p s ∗ p s ∗ − q if 1 < q < p s ∗ and r = ∞ if q ≥ p s ∗ .

Theorem 1

Suppose that conditions ( F 0 ) , ( M 1 )–( M 2 ), and ( V 1 )–( V 2 ) hold. Then, for β = 1 and 1 < q < p , there exists α ∗ > 0 such that, for any α ∈ ( 0 , α ∗ ) , equation (1) has a sequence of nontrivial solutions { u n } with J α ( u n ) < 0 , and J α ( u n ) → 0 as n → ∞ , where J α ( u ) means J α , β ( u ) with β = 1 .

Theorem 2

Suppose that conditions ( F 0 ) , ( M 1 )–( M 2 ), and ( V 1 )–( V 2 ) hold. Then, for β = 1 and q = p , there exists a positive constant m 0 ∗ such that, for each m 0 > m 0 ∗ and α ∈ ( 0 , m 0 S θ − 1 ‖ h ‖ r − 1 ) with r = p s ∗ p s ∗ − p , equation (1) has at least k pairs of nontrivial solutions, where ‖ ⋅ ‖ ν denotes the L ν -norm in R N for ν > 1 .

For the existence and multiplicity of solutions to equation (1) with θ = 2 p s , μ ∗ / p , we assume that the nonlinearity f satisfies the subcritical assumptions:

( F 1 ) f : R N × R → R is a Carathéodory function, and for any q ∈ ( p , p s ∗ ) and ε > 0 , there exists C ε > 0 such that

∣ f ( x , t ) ∣ ≤ p ε ∣ t ∣ p − 1 + q C ε ∣ t ∣ q − 1 for a.e. x ∈ R N and t ∈ R .

( F 2 ) There exists a positive exponent q 1 ∈ ( p , p s ∗ ) such that F ( x , t ) ≥ a 0 ∣ t ∣ q 1 for a.e. x ∈ R N and t ∈ R , where F ( x , t ) = ∫ 0 t f ( x , τ ) d τ .

For the Kirchhoff term M , we now turn to a typical setting as follows:

( M ) Let M ( t ) = m 0 + b t 2 p s , μ ∗ / p − 1 with b > 0 , where m 0 is the same as in condition ( M 1 ) .

Theorem 3

For β ∈ ( 0 , b S μ 2 p s , μ ∗ / p / 2 p ) , we suppose that conditions ( V 1 )–( V 2 ), ( F 1 ) –( F 2 ), and ( M ) hold. Then, there exists α ∗ > 0 such that equation (1) admits at least two distinct nontrivial solutions for α > α ∗ .

It is remarkable that Theorem 1 extends [21, Theorem 1.1] to the fractional setting, and Theorem 2 makes a generalization in the framework of fractional double-phase by adding an additional electric field V ( x ) to the original problem in [21, Theorem 1.2]. Theorem 3 is a generalization of [44] for the special case of M ( t ) = a + b t , β = 1 , and V ( x ) = 0 . It is also an extension to [21, Theorem 1.3] for another setting of s = 1 and V ( x ) = 0 .

The rest of this article, is organized as follows. In Section 2, we introduce some basic notation and several useful lemmas. We devote Section 3 to the Palais-Smale condition and then to proving Theorem 1. We present the proof of Theorem 2 in Section 4 and the proof of Theorem 3 in Section 5, respectively.

2 Notations and preliminaries

This section is dedicated to recalling basic notation regarding functional spaces and introducing some related technical lemmas, which are useful to prove our main results. Throughout this article, C ( n , ν , L , ⋯ ) stands for a universal constant depending only on prescribed quantities and possibly varies from line to line. However, the ones that we need to emphasize will be denoted by special symbols like C ν and C 0 .

Let us start by recalling the fractional Sobolev space W s , p ( R N ) :

W s , p ( R N ) = { u ∈ L p ( R N ) : [ u ] s , p < ∞ } ,

equipped with the norm

‖ u ‖ W s , p ( R N ) ≔ ( ‖ u ‖ L p ( R N ) p + [ u ] s , p p ) 1 / p

with

(3) [ u ] s , p = ∬ R 2 N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y 1 / p .

For the potential term V ( x ) , we consider the fractional Sobolev space

W V s , p ( R N ) = u ∈ W s , p ( R N ) : ∫ R N V ( x ) ∣ u ( x ) ∣ p d x < ∞

with the norm

‖ u ‖ V ≔ [ u ] s , p p + ∫ R N V ( x ) ∣ u ( x ) ∣ p d x 1 / p .

It is easy to check that ( W V s , p ( R N ) , ‖ ⋅ ‖ V ) under conditions ( V 1 ) and ( V 2 ) is a uniformly convex Banach space [9]. For any ν ∈ [ p , p s ∗ ] , the embedding W V s , p ( R N ) ↪ L ν ( R N ) is continuous [9, Theorem 6.7]. Namely, there exists a constant C ν > 0 such that

‖ u ‖ ν ≤ C ν ‖ u ‖ V , for u ∈ W V s , p ( R N ) .

Lemma 1

[37, Theorem 2.1] Assume that conditions ( V 1 ) and ( V 2 ) hold. Then, for any ν ∈ [ p , p s ∗ ) , the embedding W V s , p ↪ L ν ( R N ) is compact.

Let us define the best constant of embedding for D s , p ( R N ) → L p s ∗ ( R N ) as follows:

(4) S ≔ inf u ∈ D s , p ( R N ) \ { 0 } [ u ] s , p p ‖ u ‖ p s ∗ p > 0 ,

where D s , p ( R N ) is the so-called fractional Beppo-Levi space [37], which is the completion of C 0 ∞ ( R N ) with respect to the norm [ ⋅ ] s , p defined by (3).

Lemma 2

([22, Theorem 4.3] or [23]) Let 1 < r , t < ∞ , 0 < μ < N , and

1 r + 1 t + μ N = 2 .

Then, there exists C ( N , μ , r , t ) > 0 such that

∬ R 2 N ∣ u ( x ) ∣ ∣ ν ( y ) ∣ ∣ x − y ∣ μ d x d y ≤ C ( N , μ , r , t ) ∥ u ∥ r ∥ ν ∥ t

for u ∈ L r ( R N ) and ν ∈ L t ( R N ) .

It follows from the above Hardy-Littlewood-Sobolev inequality that there exists a constant C ˆ ( N , μ ) > 0 such that

∬ R 2 N ∣ u ( x ) ∣ p s , μ ∗ ∣ u ( y ) ∣ p s , μ ∗ ∣ x − y ∣ μ d x d y ≤ C ˆ ( N , μ ) ‖ u ‖ p s ∗ 2 p s , μ ∗ , u ∈ W V s , p ( R N ) .

Similar to (4), we define the best constant S μ by

(5) S μ ≔ inf u ∈ D s , p ( R N ) \ { 0 } [ u ] s , p p ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ d x p 2 p s , μ ∗ > 0 .

To verify that the ( P S ) condition holds for our settings, we recall the concentration-compactness principle at infinity [30]. Following [10,43], we can obtain the concentration-compactness principle for the setting of the fractional p -Laplacian as follows.

Lemma 3

Let ℳ ( R N ) denote the finite nonnegative Borel measure space on R N and { u n } n be a bounded sequence in D s , p ( R N ) satisfying

u n ⇀ u weakly in D s , p ( R N ) ; ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + s p d y ⇀ ω weakly ∗ in ℳ ( R N ) ; ∣ u n ∣ p s ∗ ⇀ ξ w e a k l y ∗ in ℳ ( R N ) ; ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ ⇀ ν weakly ∗ in ℳ ( R N ) .

We define

ω ∞ ≔ lim R → ∞ limsup n → ∞ ∫ ∣ x ∣ > R ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + s p d y d x ; ξ ∞ ≔ lim R → ∞ limsup n → ∞ ∫ ∣ x ∣ > R ∣ u n ( x ) ∣ p s ∗ d x ; ν ∞ ≔ lim R → ∞ limsup n → ∞ ∫ ∣ x ∣ > R ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ d x .

Then, we have

ω ≥ ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d y + ∑ j ∈ J ω j δ x j , ξ = ∣ u ∣ p s , μ ∗ + ∑ j ∈ J ξ j δ x j , ν = ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ + ∑ j ∈ J ν j δ x j , ξ j ≤ S − p s ∗ p ω j p s ∗ p , ν j ≤ S μ − 2 p s , μ ∗ p ω j 2 p s , μ ∗ p , j ∈ J ,

where J is at most countable, the sequences { ω j } j , { ξ j } j , { ν j } j ⊂ R 0 + , { x j } j ⊂ R N , and δ x j is the Dirac mass centered at x j . For the energy at infinity, we have

(6) limsup n → ∞ ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d y d x = ∫ R N d ω + ω ∞ ,

(7) limsup n → ∞ ∫ R N ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ d x = ∫ R N d ν + ν ∞ ,

and

(8) ξ ∞ ≤ S − p s ∗ p ω ∞ p s ∗ p , ν ∞ ≤ S μ − 2 p s , μ ∗ p ω ∞ 2 p s , μ ∗ p .

Proposition 1

([46, Lemma 2.3]) Assume that { u n } n ⊂ D s , p ( R N ) is the sequence given by Lemma 3. Let x j ∈ R N be the fixed point and ϕ ( x ) be a smooth cutoff function such that 0 ≤ ϕ ( x ) ≤ 1 , ϕ ( x ) ≡ 0 for x ∈ B 2 c ( 0 ) , ϕ ( x ) ≡ 1 for x ∈ B 1 ( 0 ) , and ∣ ∇ ϕ ( x ) ∣ ≤ 2 . Then, for any ε > 0 , we have

(9) lim ε → 0 limsup n → ∞ ∬ R 2 N ∣ ( ϕ ε , j ( x ) − ϕ ε , j ( y ) ) u n ( x ) ∣ p ∣ x − y ∣ N + p s d x d y 1 / p = 0 ,

where ϕ ε , j ( x ) = ϕ x − x j ε for x ∈ R N .

Lemma 4

([38, Theorem 2.3]) Let ( u n ) n be a bounded sequence in L p s ∗ ( R N ) such that u n → u a.e. in R N as n → ∞ . Then, we have

∫ R N ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ d x − ∫ R N ( K μ ∗ ∣ u n − u ∣ p s , μ ∗ ) ∣ u n − u ∣ p s , μ ∗ d x = ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ d x ,

as n → ∞ .

3 Proof of Theorem 1

In this section, we show that equation (1) has infinite many nontrivial solutions when β = 1 and the nonlinearity f satisfies condition ( F 0 ) for 1 < q < p . Clearly, the functional J α : W V s , p ( R N ) → R N associated with equation (1) can be expressed as follows:

J α ( u ) = 1 p M ( ‖ u ‖ V p ) − α q ∫ R N h ( x ) ∣ u ∣ q d x − 1 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ( x ) ∣ p s , μ ∗ d x .

It is a well-known fact that X ≔ W V s , p ( R N ) is a reflexive and separable Banach space, and there are { e j } j ∈ N ⊂ X and { e j ∗ } j ∈ N ⊂ X ∗ such that

(10) X = span { e j : j = 1 , 2 , … } ¯ , X ∗ = span { e j ∗ : j = 1 , 2 , … } ¯ ,

with ⟨ e i , e j ∗ ⟩ = 1 if i = j and ⟨ e i , e j ∗ ⟩ = 0 if i ≠ j .

We define

X j ≔ span { e j } , Y k = ⊕ j = 1 k X j , Z k ≔ ⊕ j = k ∞ X j ¯ ,

and let

B k ≔ { u ∈ Y k : ‖ u ‖ V ≤ ρ k } , N k ≔ { u ∈ Z k : ‖ u ‖ V = r k }

for ρ k > r k > 0 .

Definition 2

Let Φ ∈ C 1 ( X , R ) and c ∈ R . We say that the functional Φ satisfies the ( PS ) c ∗ condition if a sequence { u n } n ⊂ Y n with the property

u n ∈ Y n , Φ ( u n ) → c , Φ ∣ Y n ′ ( u n ) → 0 , as n → ∞ ,

contains a subsequence converging to a critical point of Φ .

Proposition 2

([45, dual fountain theorem]) Let Φ ∈ C 1 ( X , R ) with Φ ( − u ) = Φ ( u ) . Suppose that for every k ≥ k 0 , there exists ρ k > r k > 0 such that

  1. a k ≔ inf ‖ u ‖ = ρ k , u ∈ Z k Φ ( u ) ≥ 0 .

  2. b k ≔ max ‖ u ‖ = r k , u ∈ Y k Φ ( u ) < 0 .

  3. d k ≔ inf ‖ u ‖ ≤ ρ k , u ∈ Z k Φ ( u ) → 0 , as k → ∞ .

  4. Φ satisfies the ( PS ) c ∗ condition for every c ∈ [ d k 0 , 0 ) .

Then, Φ has a sequence of negative critical values converging to 0.

According to [45, Remarks 3.19], we know that the ( PS ) c ∗ condition implies the ( PS ) c condition.

Lemma 5

Assume that conditions ( M 1 ) and ( M 2 ) hold. If { u n } n is a ( PS ) c ∗ sequence of J α , then { u n } n is bounded in W V s , p ( R N ) .

Proof

By virtue of Hölder’s inequality for r = p s ∗ p s ∗ − q and the Sobolev embedding theorem, for all u ∈ W V s , p ( R N ) , we have

(11) ∫ R N h ( x ) ∣ u ∣ q d x ≤ S − q p ‖ h ‖ r [ u ] s , p q .

Let us fix a ( PS ) c ∗ sequence { u n } n ⊂ W V s , p ( R N ) such that J α ( u n ) → c and J α ∣ Y n ′ ( u n ) → 0 as n → ∞ . From conditions ( M 1 ) and ( M 2 ) and inequality (11), it follows that

c + o ( 1 ) ‖ u n ‖ V = J α ( u n ) − 1 2 p s , μ ∗ ⟨ J α ′ ( u n ) , u n ⟩ = 1 p M ( ‖ u n ‖ V p ) − 1 2 p s , μ ∗ M ( ‖ u n ‖ V p ) ‖ u n ‖ V p − α 1 q − 1 2 p s , μ ∗ ∫ R N h ( x ) ∣ u n ∣ q d x ≥ 1 θ p − 1 2 p s , μ ∗ M ( ‖ u n ‖ V p ) ‖ u n ‖ V p − α 1 q − 1 2 p s , μ ∗ S − q p ‖ h ‖ r [ u n ] s , p q ≥ 1 θ p − 1 2 p s , μ ∗ m 0 ‖ u n ‖ V p − α 1 q − 1 2 p s , μ ∗ S − q p ‖ h ‖ r ‖ u n ‖ V q .

This clearly indicates that { u n } n is bounded in W V s , p ( R N ) with q < p and 1 θ p − 1 2 p s , μ ∗ m 0 > 0 due to θ ∈ [ 1 , 2 p s , μ ∗ / p ) .□

Lemma 6

Assume that conditions ( M 1 ) and ( M 2 ), ( V 1 ) and ( V 2 ), and ( F 0 ) hold. Then, there exists α ∗ > 0 such that J α satisfies the ( PS ) c ∗ condition for 0 < α < α ∗ .

Proof

Let { u n } n be a ( PS ) c ∗ sequence of the functional J α , i.e.,

J α ( u n ) → c , J α ∣ Y n ′ ( u n ) → 0 , as n → ∞ .

It follows from Lemma 5 that { u n } n is bounded in W V s , p ( R N ) . Thus, there exists a function u ∈ W V s , p ( R N ) and a subsequence still denoted by { u n } n such that

u n ⇀ u weakly in W V s , p ( R N ) , u n → u in L loc t ( R N ) for t ∈ [ 1 , p s ∗ ) , u n → u a.e. in R N .

Furthermore, according to [13, Proposition 1.202], there exist the bounded nonnegative measures ω , ξ , and ν such that as n → ∞ , we have

∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d y ⇀ ω weakly ∗ in ℳ ( R N ) , ∣ u n ∣ p s ∗ ⇀ ξ weakly ∗ in ℳ ( R N ) , ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ ⇀ ν weakly ∗ in ℳ ( R N ) .

In view of Lemma 3, there exists at most countable set J , a sequence of points { x j } j ∈ J ⊂ R N , and the families of nonnegative numbers { ω j } j ∈ J , { ξ j } j ∈ J , and { ν j } j ∈ J such that

(12) ω ≥ ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d y + ∑ j ∈ J ω j δ x j , ξ = ∣ u ∣ p s ∗ + ∑ j ∈ J ξ j δ x j , ν = ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ + ∑ j ∈ J ν j δ x j

and

(13) ξ j ≤ S − p s ∗ p ω j p s ∗ p , ν j ≤ S μ − 2 p s , μ ∗ p ω j 2 p s , μ ∗ p , j ∈ J ,

where δ x j is the Dirac mass centered at x j .

We now introduce the linear functional L ( u ) on W V s , p ( R N ) defined as follows:

(14) ⟨ L ( u ) , ϕ ⟩ ≔ ∬ R 2 N ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ( ϕ ( x ) − ϕ ( y ) ) ∣ x − y ∣ N + p s d x d y + ∫ R N V ( x ) ∣ u ∣ p − 2 u ϕ d x = ⟨ u , ϕ ⟩ s , p + ∫ R N V ( x ) ∣ u ∣ p − 2 u ϕ d x

for all u ∈ W V s , p ( R N ) .

Case 1. We show that ω j = 0 for all j ∈ J . Indeed, let us construct a smooth cutoff function φ ∈ C 0 ∞ ( R N ) such that 0 ≤ φ ≤ 1 and ϕ ≡ 1 in B 1 ( 0 ) , while φ ≡ 0 in R N ⧹ B 2 ( 0 ) and ∣ ∇ φ ∣ ≤ 2 in R N . For ε > 0 , we define φ ε , j = φ ( ( x − x j ) / ε ) . Since { u n φ ε , j } is bounded in W V s , p ( R N ) , we have ⟨ J ′ ( u n ) , u n φ ε , j ⟩ → 0 as n → ∞ . That is,

(15) M ( ‖ u n ‖ V p ) ⟨ L ( u n ) , u n φ ε , j ⟩ = α ∫ R N h ( x ) ∣ u n ∣ q φ ε , j d x + ∫ R N ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ φ ε , j d x + o ( 1 ) ,

where

⟨ L ( u n ) , φ ε , j u n ⟩ = ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p φ ε , j ( x ) ∣ x − y ∣ N + p s d x d y + ∫ R N V ( x ) ∣ u n ∣ p φ ε , j d x + ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) u n ( y ) ( φ ε , j ( x ) − φ ε , j ( y ) ) ∣ x − y ∣ N + p s d x d y .

For the left-hand side of (15), using Hölder’s inequality and Lemma 3 yields

(16) lim ε → 0 limsup n → ∞ ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) ∣ x − y ∣ N + p s ( φ ε , j ( x ) − φ ε , j ( y ) ) u n ( y ) d x d y ≤ lim ε → 0 limsup n → ∞ ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y ( p − 1 ) / p ∬ R 2 N ∣ ( φ ε , j ( x ) − φ ε , j ( y ) ) u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y 1 / p ≤ C lim ε → 0 limsup n → ∞ ∬ R 2 N ∣ ( φ ε , j ( x ) − φ ε , j ( y ) ) u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y 1 / p = 0 .

This together with condition ( M 1 ) and equations (12) and (13) leads to

(17) lim ε → 0 limsup n → ∞ M ( ‖ u n ‖ V p ) ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p φ ε , j ( y ) ∣ x − y ∣ N + p s d x d y + ∫ R N V ( x ) ∣ u n ∣ p φ ε , j d x ≥ lim ε → 0 lim n → ∞ m 0 ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p φ ε , j ( y ) ∣ x − y ∣ N + p s d x d y ≥ lim ε → 0 m 0 ∬ R 2 N ∣ u ( x ) − u ( y ) ∣ p φ ε , j ( y ) ∣ x − y ∣ N + p s d x d y + ω j = m 0 ω j .

Note that { u n } is uniformly bounded in W V s , p ( R N ) and the embedding W V s , p ( R N ) ↪ L p s ∗ ( R N ) is continuous. There exists a constant M 0 > 0 independent of n such that ∫ R N ∣ u n ∣ p s ∗ d x q p s ∗ ≤ M 0 . So, for the first term on the right-hand side of (15), by Hölder’s inequality with r = p s ∗ p s ∗ − q and assumption ( F 0 ) , we obtain

(18) lim ε → 0 lim n → ∞ ∫ R N h ( x ) ∣ u n ∣ q φ ε , j d x ≤ lim ε → 0 lim n → ∞ ∫ B 2 ε ( x j ) h ( x ) ∣ u n ∣ q d x ≤ lim ε → 0 lim n → ∞ ∫ B 2 ε ( x j ) ∣ h ( x ) ∣ r d x 1 r ∫ B 2 ε ( x j ) ∣ u n ∣ p s ∗ d x q p s ∗ ≤ M 0 lim ε → 0 ∫ B 2 ε ( x j ) ∣ h ( x ) ∣ r d x 1 r = 0 .

For the second term on the right-hand side of (15), it follows from (12) that

(19) lim ε → 0 lim n → ∞ ∫ R N ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ φ ε , j d x = lim ε → 0 ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ φ ε , j d x + ν j = ν j .

Therefore, we find m 0 ω j ≤ ν j by substituting (16)–(19) into (15). Then, we obtain either ω j ≥ ( m 0 S μ 2 p s , μ ∗ p ) p 2 p s , μ ∗ − p or ω j = 0 due to (13).

To exclude the case of ω j ≥ m 0 S μ 2 p s , μ ∗ p p 2 p s , μ ∗ − p , we apply Hölder’s inequality, the Sobolev embedding, and Young’s inequality to derive

(20) α ∫ R N h ( x ) ∣ u ∣ q d x ≤ α S − q p ‖ h ‖ r ‖ u ‖ V q = α S − q p 1 θ p − 1 2 p s , μ ∗ m 0 q 1 q − 1 2 p s , μ ∗ − 1 q p ‖ u ‖ V q 1 θ p − 1 2 p s , μ ∗ m 0 q 1 q − 1 2 p s , μ ∗ − 1 − q p ‖ h ‖ r ≤ 1 θ p − 1 2 p s , μ ∗ m 0 p 1 q − 1 2 p s , μ ∗ − 1 ‖ u ‖ V p + p − q p α p p − q 1 θ p − 1 2 p s , μ ∗ − 1 q m 0 S 1 q − 1 2 p s , μ ∗ q p − q ‖ h ‖ r p p − q ,

which implies

(21) 0 > c = lim n → ∞ J α ( u n ) − 1 2 p s , μ ∗ ⟨ J α ′ ( u n ) , u n ⟩ ≥ lim n → ∞ 1 θ p − 1 2 p s , μ ∗ m 0 ‖ u n ‖ V p − 1 q − 1 2 p s , μ ∗ α ∫ R N h ( x ) ∣ u n ∣ q d x ≥ 1 θ p − 1 2 p s , μ ∗ m 0 ( ‖ u ‖ V p + ω j ) − 1 q − 1 2 p s , μ ∗ α ∫ R N h ( x ) ∣ u ∣ q d x ≥ 1 θ p − 1 2 p s , μ ∗ m 0 ω j − p − q p α p p − q 1 θ p − 1 2 p s , μ ∗ − 1 q m 0 S 1 q − 1 2 p s , μ ∗ q p − q ‖ h ‖ r p p − q ≥ 1 θ p − 1 2 p s , μ ∗ ( m 0 S μ ) 2 p s , μ ∗ 2 p s , μ ∗ − p − p − q p α p p − q 1 θ p − 1 2 p s , μ ∗ − 1 q m 0 S 1 q − 1 2 p s , μ ∗ q p − q ‖ h ‖ r p p − q .

Therefore, we can choose α 1 > 0 so small that, for every α ∈ ( 0 , α 1 ) , the right-hand side of (21) is greater than zero, which leads to a contradiction.

Case 2. To rule out the possibility of concentration for mass at infinity, we take a suitable cutoff function ψ R ∈ C 0 ∞ ( 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 . In view of the definitions of ω ∞ and ν ∞ given in Lemma 3, we obtain

(22) ω ∞ = lim R → ∞ limsup n → ∞ ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p ψ R ( y ) ∣ x − y ∣ N + p s d x d y

and

(23) ν ∞ = lim R → ∞ limsup n → ∞ ∫ R N ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ ψ R d x .

Then, the fact ⟨ J ′ ( u n ) , u n ψ R ⟩ → 0 as n → ∞ implies that

(24) M ( ‖ u n ‖ V p ) ⟨ L ( u n ) , u n ψ R ⟩ = α ∫ R N h ( x ) ∣ u n ∣ q ψ R d x + ∫ R N ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ ψ R d x + o ( 1 ) ,

where

⟨ L ( u n ) , ψ R u n ⟩ = ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p ψ R ( x ) ∣ x − y ∣ N + p s d x d y + ∫ R N V ( x ) ∣ u n ∣ p ψ R d x + ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) u n ( y ) ( ψ R ( x ) − ψ R ( y ) ) ∣ x − y ∣ N + p s d x d y .

For the left-hand side of (24), it follows from Hölder’s inequality that

(25) lim R → ∞ limsup n → ∞ ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) ∣ x − y ∣ N + p s ( ψ R ( x ) − ψ R ( y ) ) u n ( y ) d x d y ≤ lim R → ∞ limsup n → ∞ ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y ( p − 1 ) / p ∬ R 2 N ∣ ( ψ R ( x ) − ψ R ( y ) ) u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y 1 / p ≤ C lim R → ∞ limsup n → ∞ ∬ R 2 N ∣ ( ψ R ( x ) − ψ R ( y ) ) u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y 1 / p .

Using an argument analogous to the proof of (9) and [46, (2.15)], we can obtain

lim R → ∞ limsup n → ∞ ∬ R 2 N ∣ ( ψ R ( x ) − ψ R ( y ) ) u n ( x ) ∣ p ∣ x − y ∣ N + p s d x d y 1 / p = 0 .

Under condition ( M 1 ) , it follows from (22) and (24) that

(26) lim R → ∞ limsup n → ∞ M ( ‖ u n ‖ V p ) ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p ψ R ( y ) ∣ x − y ∣ N + p s d x d y + ∫ R N V ( x ) ∣ u n ∣ p ψ R d x ≥ lim R → ∞ lim n → ∞ m 0 ∬ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p ψ R ( y ) ∣ x − y ∣ N + p s d x d y = m 0 ω ∞ .

For the first term on the right-hand side of (24), using Hölder’s inequality and condition ( F 0 ) leads to

(27) lim R → ∞ lim n → ∞ ∫ R N h ( x ) ∣ u n ∣ q ψ R d x ≤ lim R → ∞ lim n → ∞ ∫ ∣ x ∣ > 2 R h ( x ) ∣ u n ∣ q d x ≤ lim R → ∞ lim n → ∞ ∫ ∣ x ∣ > 2 R ∣ h ( x ) ∣ r d x 1 / r ∫ ∣ x ∣ > 2 R ∣ u n ∣ p s ∗ d x q / p s ∗ ≤ M 0 lim R → ∞ ∫ ∣ x ∣ > 2 R ∣ h ( x ) ∣ r d x 1 / r = 0 .

By substituting (23) and (25)–(27) into (24), we derive m 0 ω ∞ ≤ ν ∞ . So we obtain either ω ∞ ≥ ( m 0 S μ 2 p s , μ ∗ p ) p 2 p s , μ ∗ − p or ω ∞ = 0 . To exclude the possibility of the case of ω ∞ ≥ ( m 0 S μ 2 p s , μ ∗ p ) p 2 p s , μ ∗ − p , as discussing for (20) and (21), we can deduce

0 > c ≥ 1 θ p − 1 2 p s , μ ∗ ( m 0 S μ ) 2 p s , μ ∗ 2 p s , μ ∗ − p − p − q p α p p − q 1 θ p − 1 2 p s , μ ∗ − 1 q m 0 S 1 q − 1 2 p s , μ ∗ q p − q ‖ h ( x ) ‖ r p p − q .

Therefore, if we take α 2 > 0 so small that α ∈ ( 0 , α 2 ) , then the right-hand side of the above inequality is greater than zero. This leads to a contradiction.

Combining Cases 1 and 2, for any c < 0 and α ∈ ( 0 , α ∗ ) with α ∗ ≔ min { α 1 , α 2 } , we obtain

(28) ω j = 0 , for j ∈ J and ω ∞ = 0 .

As n → ∞ , there holds

∫ R N ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ d x → ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ d x .

It follows from Lemma 4 that

(29) ∫ R N ( K μ ∗ ∣ u n − u ∣ p s , μ ∗ ) ∣ u n − u ∣ p s , μ ∗ d x → 0 , as n → ∞ .

Using (28), Hölder’s inequality and the Brezis-Lieb lemma, we derive

(30) ∫ R N h ( x ) ∣ u n ∣ q − 2 u n ( u n − u ) d x ≤ ‖ h ‖ r ‖ u n ‖ p s ∗ q − 1 ‖ u n − u ‖ p s ∗ → 0 , as n → ∞ ,

and

(31) ∫ R N h ( x ) ∣ u ∣ q − 2 u ( u n − u ) d x ≤ ‖ h ‖ r ‖ u ‖ p s ∗ q − 1 ‖ u n − u ‖ p s ∗ → 0 , as n → ∞ .

We are now in a position to prove that { u n } converges strongly to u in W V s , p ( R N ) . To this end, we start with proving the identity

(32) M ( ‖ u n ‖ V p ) ⟨ L ( u n ) , u n − u ⟩ − M ( ‖ u ‖ V p ) ⟨ L ( u ) , u n − u ⟩ = M ( ‖ u n ‖ V p ) ⟨ L ( u n ) − L ( u ) , u n − u ⟩ .

Indeed, given that L ( u ) is the continuous linear functional in W V s , p ( R N ) , the weak convergence of { u n } n in W V s , p ( R N ) implies that

lim n → ∞ ⟨ L ( u ) , u n − u ⟩ = 0 .

Note that { u n } is uniformly bounded and M ( ⋅ ) ∈ C ( R 0 + ) . So we see that both M ( ‖ u n ‖ V p ) and M ( ‖ u ‖ V p ) are uniformly bounded, which leads to lim n → ∞ M ( ‖ u n ‖ V p ) ⟨ L ( u ) , u n − u ⟩ = lim n → ∞ M ( ‖ u ‖ V p ) ⟨ L ( u ) , u n − u ⟩ = 0 . Therefore, as n → ∞ , we obtain

M ( ‖ u n ‖ V p ) ⟨ L ( u n ) , u n − u ⟩ − M ( ‖ u ‖ V p ) ⟨ L ( u ) , u n − u ⟩ = M ( ‖ u n ‖ V p ) ⟨ L ( u n ) − L ( u ) , u n − u ⟩ + M ( ‖ u n ‖ V p ) − M ( ‖ u ‖ V p ) ⟨ L ( u ) , u n − u ⟩ = M ( ‖ u n ‖ V p ) ⟨ L ( u n ) − L ( u ) , u n − u ⟩ .

Clearly, ⟨ J α ′ ( u n ) − J α ′ ( u ) , u n − u ⟩ → 0 as n → ∞ . Hence, using (29)–(32) yields that

(33) o n ( 1 ) = ⟨ J α ′ ( u n ) − J α ′ ( u ) , u n − u ⟩ = M ( ‖ u n ‖ V p ) ⟨ L ( u n ) , u n − u ⟩ − M ( ‖ u ‖ V p ) ⟨ L ( u ) , u n − u ⟩ − α ∫ R N h ( x ) ( ∣ u n ∣ q − 2 u n − ∣ u ∣ q − 2 u ) ( u n − u ) d x − ∫ R N ( ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ − 2 u n ( u n − u ) − ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ − 2 u ( u n − u ) ) d x = M ( ‖ u n ‖ V p ) ⟨ L ( u n ) − L ( u ) , u n − u ⟩ − ∫ R N ( K μ ∗ ∣ u n − u ∣ p s , μ ∗ ) ∣ u n − u ∣ p s , μ ∗ d x = M ( ‖ u n ‖ V p ) ⟨ L ( u n ) − L ( u ) , u n − u ⟩ .

By recalling the so-called Simon inequality [20], for ξ , η ∈ R , there exists a constant C = C ( p , N ) > 0 such that

(34) ∣ ξ − η ∣ p ≤ C ( ∣ ξ ∣ p − 2 ξ − ∣ η ∣ p − 2 η ) ( ξ − η ) , for p ≥ 2 , C [ ( ∣ ξ ∣ p − 2 ξ − ∣ η ∣ p − 2 η ) ( ξ − η ) ] p / 2 ( ∣ ξ ∣ p + ∣ η ∣ p ) ( 2 − p ) / 2 , for 1 < p < 2 .

Let ξ = u n ( x ) − u n ( y ) and η = u ( x ) − u ( y ) . Using condition ( M 1 ) , we have

(35) o n ( 1 ) = ( ⟨ L ( u n ) , u n − u ⟩ − ⟨ L ( u ) , u n − u ⟩ ) = ⟨ u n , u n − u ⟩ s , p − ⟨ u , u n − u ⟩ s , p + ∫ R N V ( x ) ( ∣ u n ∣ p − 2 u n − ∣ u ∣ p − 2 u ) ( u n − u ) d x ≔ A 1 + A 2

and

⟨ u , v ⟩ s , p ≔ ∬ R 2 N ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ( v ( x ) − v ( y ) ) ∣ x − y ∣ N + s p d x d y ,

where

A 1 = ⟨ u n , u n − u ⟩ s , p − ⟨ u , u n − u ⟩ s , p , A 2 = ∫ R N V ( x ) ( ∣ u n ∣ p − 2 u n − ∣ u ∣ p − 2 u ) ( u n − u ) d x .

For the case of p ≥ 2 , we have

[ u n − u ] s , p p = ∬ R 2 N ∣ ( u n ( x ) − u n ( y ) ) − ( u ( x ) − u ( y ) ) ∣ p ∣ x − y ∣ N + p s d x d y ≤ C ( ⟨ u n , u n − u ⟩ s , p − ⟨ u , u n − u ⟩ s , p ) = A 1

and

∫ R N V ( x ) ∣ u n − u ∣ p d x ≤ C ∫ R N V ( x ) ( ∣ u n ∣ p − 2 u n − ∣ u ∣ p − 2 u ) ( u n − u ) d x = A 2 .

That is, ‖ u n − u ‖ V = o n ( 1 ) for p ≥ 2 .

For the case of 1 < p < 2 , it follows from (34) and Hölder’s inequality that

(36) [ u n − u ] s , p p = ∬ R 2 N ∣ ( u n ( x ) − u n ( y ) ) − ( u ( x ) − u ( y ) ) ∣ p ∣ x − y ∣ N + p s d x d y ≤ C 1 ( ⟨ u n , u n − u ⟩ s , p − ⟨ u , u n − u ⟩ s , p ) p / 2 ( [ u n ] s , p p ( 2 − p ) / 2 + [ u ] s , p p ( 2 − p ) / 2 ) ≤ C 2 ( ⟨ u n , u n − u ⟩ s , p − ⟨ u , u n − u ⟩ s , p ) p / 2 = A 1 p 2 ,

where we have used the inequality

( a + b ) s ≤ a s + b s , for a , b > 0 and s ∈ ( 0 , 1 ) .

Similarly, for 1 < p < 2 , from (34) it follows that

(37) ∫ R N V ( x ) ∣ u n − u ∣ p d x ≤ C ∫ R N V ( x ) [ ( ∣ u n ∣ p − 2 u n − ∣ u ∣ p − 2 u ) ( u n − u ) ] p / 2 ( ∣ u n ∣ p + ∣ u ∣ p ) ( 2 − p ) / 2 d x ≤ C 1 ∫ R N V ( x ) ( ∣ u n ∣ p − 2 u n − ∣ u ∣ p − 2 u ) ( u n − u ) d x p / 2 ∫ R N V ( x ) ( ∣ u n ∣ p + ∣ u ∣ p ) d x ( 2 − p ) / 2 ≤ C 2 ∫ R N V ( x ) ( ∣ u n ∣ p − 2 u n − ∣ u ∣ p − 2 u ) ( u n − u ) d x p / 2 = A 2 p 2 .

By combining (36) and (37), we arrive at ‖ u n − u ‖ V = o n ( 1 ) for 1 < p < 2 .□

Lemma 7

Assume that conditions ( V 1 ) and ( V 2 ) hold. Then, for 1 ≤ q 0 < p s ∗ , we have

β k ≔ sup u ∈ Z k ‖ u ‖ V = 1 ‖ u ‖ q 0 → 0 , a s k → ∞ .

Proof

From Lemma 1, we know that W V s , p ( R N ) ↪ L ν ( R N ) is compact for 1 ≤ ν < p s ∗ . Therefore, we can take a sequence 0 < β k + 1 ≤ β k < ∞ such that β k → β 0 ≥ 0 as k → ∞ . For k ≥ 1 , there exists u k ∈ Z k such that ‖ u k ‖ V = 1 and ‖ u k ‖ q 0 > β k / 2 . On the other hand, from the definition of Z k , we have u k ⇀ 0 as k → ∞ in W V s , p ( R N ) . The Sobolev compact imbedding theorem implies that u k → 0 as k → ∞ in L q 0 ( R N ) . Thus, we have β 0 = 0 .□

We are now ready to prove Theorem 1.

Proof of Theorem 1

In view of Proposition 2 and Lemma 6, we only need to verify the assumptions ( A 1 )–( A 3 ).

For ( A 1 ) , due to 2 p s , μ ∗ > p , we may choose R > 0 sufficiently small such that

‖ u ‖ V ≤ R ⇒ 1 2 p s , μ ∗ ( S μ − 1 ‖ u ‖ V p ) 2 p s , μ ∗ p ≤ m 0 2 θ p ‖ u ‖ V p .

For u ∈ Z k with ‖ u ‖ V ≤ R , from the definition of S μ and conditions ( M 1 ) and ( F 0 ) , it follows that

(38) J α ( u ) ≥ 1 θ p m 0 ‖ u ‖ V p − α β k q q ‖ h ‖ r ‖ u ‖ V q − 1 2 p s , μ ∗ ( S μ − 1 ‖ u ‖ V p ) 2 p s , μ ∗ p ≥ m 0 2 θ p ‖ u ‖ V p − α β k q q ‖ h ‖ r ‖ u ‖ V q = m 0 2 θ p ‖ u ‖ V p − q − α β k q q ‖ h ‖ r ‖ u ‖ V q .

Let ρ k ≔ 2 θ p m 0 ‖ h ‖ r α β k q q 1 / ( p − q ) . Since β k → 0 as k → ∞ due to Lemma 7, we have ρ k → 0 as k → ∞ . So there exists a positive constant k 0 such that ρ k ≤ R while k ≥ k 0 . Therefore, for k ≥ k 0 , u ∈ Z k , and ‖ u ‖ V = ρ k , we have J α ( u ) ≥ 0 to ensure condition ( A 1 ) .

For ( A 2 ) , for any u ∈ Y k and ‖ u ‖ V = r k < 1 with 0 < r k < ρ k , we obtain

J α ( u ) = 1 p M ( ‖ u ‖ V p ) − α q ∫ R N h ( x ) ∣ u ∣ q d x − 1 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u ( x ) ∣ p s , μ ∗ ) ∣ u ( x ) ∣ p s , μ ∗ d x ≤ 1 p max 0 < t < 1 M ( t ) ‖ u ‖ V p − α q ‖ u ‖ V q − S μ − 2 p s , μ ∗ / p 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ .

Given the equivalence of the norms in the finite dimensional space Y k , we know that condition ( A 2 ) is true for sufficiently small r k > 0 while α > 0 .

For ( A 3 ) , for k ≥ k 0 , u ∈ Z k , and ‖ u ‖ V ≤ ρ k , it follows from (38) that

J α ( u ) ≥ − α β k q p ‖ h ‖ r ‖ u ‖ V q ≥ − α β k q p ‖ h ‖ r ρ k q .

In view of β k → 0 and ρ k → 0 as k → ∞ , we see that condition ( A 3 ) is satisfied.□

4 Proof of Theorem 2

In this section, we apply the mountain pass theorem for even functionals to prove the multiplicity of the solutions for equation (1) under condition ( F 0 ) with q = p and β = 1 . We rewrite equation (1) as follows:

(39) M ( ‖ u ‖ V p ) [ ( − Δ ) p s u + V ( x ) ∣ u ∣ p − 2 u ] = α h ( x ) ∣ u ∣ p − 2 u + ∫ R N ∣ u ( y ) ∣ p s , μ ∗ ∣ x − y ∣ μ d y ∣ u ∣ p s , μ ∗ − 2 u , in R N .

So the associated functional J α with equation (39) is

J α ( u ) = 1 p M ( ‖ u ‖ V p ) − α p ∫ R N h ( x ) ∣ u ∣ p d x − 1 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ d x .

Lemma 8

Assume that conditions ( M 1 ) and ( M 2 ) and ( V 1 ) and ( V 2 ) are satisfied. For α ∈ ( 0 , m 0 S θ − 1 ‖ h ‖ r − 1 ) with θ being defined in condition ( M 2 ) , we suppose that { u n } n is a ( PS ) c sequence of J α in W V s , p ( R N ) satisfying

(40) c < c ∗ w i t h c ∗ ≔ 1 θ p − 1 2 p s , μ ∗ ( m 0 S μ ) 2 p s , μ ∗ 2 p s , μ ∗ − p .

Then, { u n } n contains a strongly convergent subsequence.

Proof

Using Hölder’s inequality for r = p s ∗ p s ∗ − p and the Sobolev embedding theorem yields

(41) ∫ R N h ( x ) ∣ u ∣ p d x ≤ S − 1 ‖ h ( x ) ‖ r ‖ u ‖ V p

for u ∈ W V s , p ( R N ) .

Let us fix a ( PS ) c sequence { u n } n for J α in W V s , p ( R N ) at the level c < c ∗ . Given the fact α ∈ ( 0 , m 0 S θ − 1 ‖ h ‖ r − 1 ) and (41), by an argument similar to the proof of Lemma 6 and (21), we deduce

c ∗ > c = lim n → ∞ J α ( u n ) − 1 2 p s , μ ∗ ⟨ J α ′ ( u n ) , u n ⟩ ≥ 1 θ p − 1 2 p s , μ ∗ m 0 ( ‖ u ‖ V p + ω j ) − 1 p − 1 2 p s , μ ∗ α S − 1 ‖ h ‖ r ‖ u ‖ V p = 1 θ p − 1 2 p s , μ ∗ m 0 ω j + 1 θ p − 1 2 p s , μ ∗ m 0 − 1 p − 1 2 p s , μ ∗ α S − 1 ‖ h ‖ r ‖ u ‖ V p ≥ 1 θ p − 1 2 p s , μ ∗ m 0 ω j ≥ 1 θ p − 1 2 p s , μ ∗ ( m 0 S μ ) 2 p s , μ ∗ 2 p s , μ ∗ − p = c ∗ .

Clearly, this leads a contradiction. Therefore, we obtain the compactness of the ( PS ) c sequence with c < c ∗ .□

Let us now recall a version of the mountain pass theorem regarding even functionals and the Krasnoselskii’s genus theory [39], which plays an important role to prove Theorem 2.

Proposition 3

Let X be an infinite dimensional Banach space with X = V ⊕ Y , where V is a finite k dimensional subspace with V = span { e 1 , e 2 , … , e k } . Suppose that J ∈ C 1 ( X ) is an even functional with J ( 0 ) = 0 , which satisfies the following three conditions:

  1. There exist the constants ϱ , ρ > 0 such that J ( u ) ≥ ϱ for u ∈ ∂ B ρ ( 0 ) ∩ Y .

  2. There exists a constant c ∗ > 0 such that J satisfies the ( PS ) c condition for c ∈ ( 0 , c ∗ ) .

  3. For each finite dimensional subspace X ˆ ⊂ X , there exists R = R ( X ˆ ) such that J ( u ) ≤ 0 for u ∈ X ˆ \ B R ( 0 ) .

For n ≥ k , we inductively choose e n + 1 ∉ X n ≔ span { e 1 , e 2 , … , e n } and set

R n = R ( X n ) , D n = B R n ( 0 ) ∩ X n , Σ ≔ { E : E i s c l o s e d i n X a n d s y m m e t r i c w i t h r e s p e c t t o t h e o r i g i n } , G n ≔ { h ∈ C ( D n , X ) : h i s o d d a n d h ( u ) = u f o r a l l u ∈ ∂ B R n ( 0 ) ∩ X n }

and

(42) Γ j ≔ { h ( D n \ E ¯ ) : h ∈ G n , n ≥ j , E ∈ Σ a n d γ ( E ) ≤ n − j } ,

where γ ( E ) is Krasnoselskii’s genus of E. For each j ∈ N , let c j ≔ inf B ∈ Γ j max u ∈ B J ( u ) . Then, for c j < c ∗ and 0 < ϱ ≤ c j ≤ c j + 1 for j > k , c j is a critical value of J. Moreover, if c j = c j + 1 = … = c j + l = c < c ∗ for j > k , then we have γ ( K c ) ≥ l + 1 , where

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

Lemma 9

Under the hypotheses described in Theorem 2  for any α ∈ ( 0 , m 0 S θ − 1 ‖ h ‖ r − 1 ) , the functional J α satisfies conditions ( I 1 )–( I 3 ).

Proof

For ( I 1 ) , in view of condition ( M 1 ) and definitions of S and S μ , we obtain

J α ( u ) ≥ 1 θ p m 0 ‖ u ‖ V p − α p S − 1 ‖ h ‖ r ‖ u ‖ V p − S μ − 2 p s , μ ∗ p 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ = 1 p m 0 θ − α S ‖ h ‖ r ‖ u ‖ V p − S μ − 2 p s , μ ∗ p 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ .

Note that p < 2 p s , μ ∗ and m 0 θ − α S ‖ h ‖ r > 0 due to α ∈ ( 0 , m 0 S θ − 1 ‖ h ‖ r − 1 ) . There exists a ϱ > 0 satisfying J α ( u ) ≥ ϱ > 0 for u ∈ W V s , p ( R N ) with ‖ u ‖ V = ρ such that ρ > 0 sufficiently small. This ensures that J α satisfies condition ( I 1 ) .

For ( I 2 ) , from Lemma 8, it is easy to check that ( I 2 ) is true for α ∈ ( 0 , m 0 S θ − 1 ‖ h ‖ r − 1 ) and

c ∗ = 1 θ p − 1 2 p s , μ ∗ ( m 0 S μ ) 2 p s , μ ∗ 2 p s , μ ∗ − p .

For ( I 3 ) , from condition ( M 2 ) and M ( t ) = ∫ 0 t M ( τ ) d τ , it follows that

(43) M ( t ) ≤ M ( 1 ) t θ , t ∈ [ 1 , ∞ ) .

Let E be a finite dimensional subspace of W V s , p ( R N ) . For any u ∈ E with ‖ u ‖ V = R > 1 it follows from (43) that

J α ( u ) ≤ 1 p M ( ‖ u ‖ V p ) − α p ∫ R N h ( x ) ∣ u ∣ p d x − 1 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ d x ≤ 1 p M ( 1 ) ‖ u ‖ V θ p − S μ − 2 p s , μ ∗ p 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ .

Since θ p < 2 p s , μ ∗ , we obtain J α ( u ) < 0 for u ∈ E with ‖ u ‖ V = R , where R is chosen large enough. Consequently, J α satisfies condition ( I 3 ) as desired.□

Lemma 10

There exists a sequence { M n } n ⊂ R + independent of α such that M n ≤ M n + 1 for all n ∈ N and for any α > 0 , there holds

c n α ≔ inf B ∈ Γ n max u ∈ B J α ( u ) < M n ,

where Γ n is defined by (42).

Proof

From the definition of c n α and the fact of h ( x ) ≥ 0 with h ( x ) ≢ 0 in R N , we deduce

c n α = inf B ∈ Γ n max u ∈ B 1 p M ( ‖ u ‖ V p ) − α p ∫ R N h ( x ) ∣ u ∣ p d x − 1 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ d x < inf B ∈ Γ n max u ∈ B 1 p M ( ‖ u ‖ V p ) − 1 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ d x ≔ M n .

Thus, we obtain M n < ∞ and M n ≤ M n + 1 based on the definition of Γ n .□

Proof of Theorem 2

According to Lemma 10, we can choose m 0 ∗ > 0 so large that, for any m 0 > m 0 ∗ , there holds

M k < 1 θ p − 1 2 p s , μ ∗ ( m 0 S μ ) 2 p s , μ ∗ 2 p s , μ ∗ − p = c ∗ ,

which yields

c k α < M k < 1 θ p − 1 2 p s , μ ∗ ( m 0 S μ ) 2 p s , μ ∗ 2 p s , μ ∗ − p .

Then, for all α ∈ ( 0 , m 0 S θ − 1 ‖ h ‖ r − 1 ) , we obtain

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

It follows from Proposition 3 that each of the levels 0 < c 1 α ≤ c 2 α ≤ … ≤ c k α is a critical value of J α . If c 1 α < c 2 α < … < c k α is true, then the functional J α has at least k pairs of critical points. Otherwise, it gives rise to c j α = c j + 1 α for some j = 1 , 2 , … , k − 1 . Using Proposition 3 again implies that K c j α is infinite set (cf. [39, Chapter 7]). Then equation (39) has infinitely many weak solutions in this case. Consequently, equation (1) has at least k pairs of nontrivial solutions.□

5 Proof of Theorem 3

Under the conditions described in Theorem 3, we rewrite equation (1) as follows:

(44) ( m 0 + b ‖ u ‖ V ( 2 p s , μ ∗ / p − 1 ) p ) [ ( − Δ ) p s u + V ( x ) ∣ u ∣ p − 2 u ] = α f ( x , u ) + β ∫ R N ∣ u ( y ) ∣ p s , μ ∗ ∣ x − y ∣ μ d y ∣ u ∣ p s , μ ∗ − 2 u

with the associated functional J α , β : W V s , p ( R N ) → R N :

(45) J α , β ( u ) ≔ m 0 p ‖ u ‖ V p + b 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ − α ∫ R N F ( x , u ) d x − β 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u ( x ) ∣ p s , μ ∗ ) ∣ u ( x ) ∣ p s , μ ∗ d x .

Lemma 11

For 0 < β < b S μ 2 p s , μ ∗ / p / 2 p , the functional J α , β satisfies the (PS) condition in W V s , p ( R N ) for α > 0 .

Proof

Let { u n } n ⊂ W V s , p ( R N ) be a ( PS ) c sequence of the functional J α , β , i.e.,

J α , β ( u n ) → c , J α , β ′ ( u n ) → 0 as n → ∞ .

From condition ( F 1 ) , we have

∣ F ( x , u n ) ∣ ≤ ε ∣ u n ∣ p + C ε ∣ u n ∣ q for a.e. x ∈ R N .

For u n ∈ W V s , p ( R N ) , we deduce

c + o ( 1 ) = J α , β ( u n ) ≥ b 2 p s , μ ∗ ‖ u n ‖ V 2 p s , μ ∗ − α ∫ R N F ( x , u n ) d x − β 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ d x ≥ b 2 p s , μ ∗ ‖ u n ‖ V 2 p s , μ ∗ − α ε C p p ‖ u n ‖ V p − α C ε C q q ‖ u n ‖ V q − β 2 p s , μ ∗ S μ − 2 p s , μ ∗ p ‖ u n ‖ V 2 p s , μ ∗ = 1 2 p s , μ ∗ b − β S μ − 2 p s , μ ∗ p ‖ u n ‖ V 2 p s , μ ∗ − α ε C p p ‖ u n ‖ V p − C ε C q q ‖ u n ‖ V q .

We note that J α , β ( u n ) is coercive and bounded from below in W V s , p ( R N ) due to p < q < p s ∗ < 2 p s , μ ∗ for μ < N and 0 < β < b S 2 p s , μ ∗ / p . This implies that { u n } n is uniformly bounded in W V s , p ( R N ) . So there exists u ∈ W V s , p ( R N ) and a subsequence of { u n } n , still denoted by { u n } n , such that

u n ⇀ u in W V s , p ( R N ) and in L p s ∗ ( R N ) , u n → u a.e. in R N , u n → u in L loc τ ( R N ) for 1 ≤ τ < p s ∗ , ∣ u n ∣ p s ∗ − 2 u n ⇀ ∣ u ∣ p s ∗ − 2 u in L p s ∗ p s ∗ − 1 ( R N ) .

For u ∈ W V s , p ( R N ) , it follows from Hölder’s inequality that

∣ ⟨ L ( u ) , ν ⟩ ∣ ≔ ∬ R 2 N ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ( ν ( x ) − ν ( y ) ) ∣ x − y ∣ N + p s d x d y + ∫ R N V ( x ) ∣ u ∣ p − 2 u ν d x ≤ [ u ] s , p p − 1 [ ν ] s , p + ∫ R N V ( x ) ∣ u ∣ p d x ( p − 1 ) / p ∫ R N V ( x ) ∣ ν ∣ p d x 1 / p ≤ [ u ] s , p p − 1 + ∫ R N V ( x ) ∣ u ∣ p d x ( p − 1 ) / p ‖ ν ‖ V ,

where L ( u ) is defined by (14). Therefore, the linear functional L ( u ) is continuous in W V s , p ( R N ) . Form the weak convergence of { u n } n in W V s , p ( R N ) , we obtain

(46) lim n → ∞ ⟨ L ( u ) , u n − u ⟩ = 0 .

We now show that

(47) lim n → ∞ ∫ R n ( f ( x , u n ) − f ( x , u ) ) ( u n − u ) d x = 0 .

In fact, up to a subsequence, we see that u n → u in L γ as n → ∞ for γ = p or q ∈ [ p , p s ∗ ) due to the Sobolev compact embedding (see Lemma 1). Note that

∣ f ( x , t ) ∣ ≤ ∣ t ∣ p − 1 + C ε ∣ t ∣ q − 1 for a.e. ( x , t ) ∈ R N × R .

It follows from Hölder’s inequality that

∫ R N ( f ( x , u n ) − f ( x , u ) ) ( u n − u ) d x ≤ ∫ R N ( ( ∣ u n ∣ p − 1 + ∣ u ∣ p − 1 ) ∣ u n − u ∣ + C ε ( ∣ u n ∣ q − 1 + ∣ u ∣ q − 1 ) ∣ u n − u ∣ ) d x ≤ ( ‖ u n ‖ p p − 1 + ‖ u ‖ p p − 1 ) ‖ u n − u ‖ p + C ε ( ‖ u n ‖ q q − 1 + ‖ u ‖ q q − 1 ) ‖ u n − u ‖ q .

This explicitly implies (47).

Given that { u n } n is a ( PS ) c sequence, from Lemma 4 and (46), we derive

(48) o n ( 1 ) = ⟨ J α , β ′ ( u n ) − J α , β ′ ( u ) , u n − u ⟩ = ( m 0 + b ‖ u n ‖ V ( θ − 1 ) p ) ( ⟨ L ( u n ) , u n − u ⟩ − ⟨ L ( u ) , u n − u ⟩ ) − α ∫ R N ( f ( x , u n ) u n − f ( x , u ) u ) d x − β ∫ R N ( ( K μ ∗ ∣ u n ∣ p s , μ ∗ ) ∣ u n ∣ p s , μ ∗ − 2 u n ( u n − u ) − ( K μ ∗ ∣ u ∣ p s , μ ∗ ) ∣ u ∣ p s , μ ∗ − 2 u ( u n − u ) ) d x = ( m 0 + b ‖ u n ‖ V ( θ − 1 ) p ) ⟨ L ( u n ) − L ( u ) , u n − u ⟩ − β ∫ R N ( K μ ∗ ∣ u n − u ∣ p s , μ ∗ ) ∣ u n − u ∣ p s , μ ∗ d x .

According to [40, Lemma 3.2], we find that

(49) [ u n ] s , p p = [ u n − u ] s , p p + [ u ] s , p p + o n ( 1 ) .

Employing the Brezis-Lieb lemma yields

(50) ∫ R N V ( x ) ∣ u n − u ∣ p d x = ∫ R N V ( x ) ∣ u n ∣ p d x − ∫ R N V ( x ) ∣ u ∣ p d x + o ( 1 ) .

It follows from (48)–(50) that

(51) ( m 0 + b ( ‖ u n − u ‖ V + ‖ u ‖ V ) ( 2 p s , μ ∗ / p − 1 ) p ) ⟨ L ( u n ) − L ( u ) , u n − u ⟩ = β ∫ R N ( K μ ∗ ∣ u n − u ∣ p s , μ ∗ ) ∣ u n − u ∣ p s , μ ∗ d x + o ( 1 ) .

Using the Simon inequality (34) with p ≥ 2 leads to

(52) ⟨ L ( u n ) − L ( u ) , u n − u ⟩ ≥ 1 2 p ‖ u n − u ‖ V p .

In view of the definition of S μ , we obtain

∫ R N ( K μ ∗ ∣ u n − u ∣ p s , μ ∗ ) ∣ u n − u ∣ p s , μ ∗ d x ≤ S μ − 2 p s , μ ∗ p ‖ u n − u ‖ V 2 p s , μ ∗ .

From (51) and (52), it follows that

1 2 p ( m 0 + b ( ‖ u n − u ‖ V + ‖ u ‖ V ) ( θ − 1 ) p ) ‖ u n − u ‖ V p ≤ β S μ − 2 p s , μ ∗ p ‖ u n − u ‖ V 2 p s , μ ∗ .

Let lim n → ∞ ‖ u n − u ‖ V = η . Then

(53) b 2 p η 2 p s , μ ∗ ≤ β S μ − 2 p s , μ ∗ p η 2 p s , μ ∗ .

Note that β < b S μ 2 p s , μ ∗ p / 2 p . It means η = 0 . Consequently, we arrive at the desired result.□

Proof of Theorem 3

Let us first show that the functional (45) has a nontrivial least energy solution. Note that the functional J α , β ( u ) is coercive, continuous, and bounded from below in W V s , p ( R N ) . According to Lemma 11, there exists a global minimizer u 1 ∈ W V s , p ( R N ) such that

J α , β ( u 1 ) = m ≔ inf u ∈ W V s , p ( R N ) J α , β ( u ) .

Take a function e ∈ W V s , p ( R N ) with ‖ e ‖ V = 1 . It follows from condition ( F 2 ) that

(54) J α , β ( e ) = m 0 p + b 2 p s , μ ∗ − α ∫ R N F ( x , e ) d x − β 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ e ( x ) ∣ p s , μ ∗ ) ∣ e ( x ) ∣ p s , μ ∗ d x ≤ m 0 p + b 2 p s , μ ∗ − α ∫ R N F ( x , e ) d x ≤ m 0 p + b 2 p s , μ ∗ − α a 0 ‖ e ‖ q 1 q 1 < 0

for α > α ∗ with α ∗ = m 0 p + b 2 p s , μ ∗ / ( a 0 ‖ e ‖ q 1 q 1 ) . Therefore, u 1 is a nontrivial least energy solution of (45) satisfying J α , β ( u 1 ) = m < 0 .

We now prove that equation (45) has a mountain pass solution too. According to condition ( F 1 ) , for any ε > 0 , there exists C ε > 0 such that

∣ F ( x , t ) ∣ ≤ ε ∣ t ∣ p + C ε ∣ t ∣ q for a.e. x ∈ R N and t ∈ R .

In view of the definition of S μ , for any u ∈ W V s , p ( R N ) , we deduce

J α , β ( u ) = m 0 p ‖ u ‖ V p + b 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ − α ∫ R N F ( x , u ) d x − β 2 p s , μ ∗ ∫ R N ( K μ ∗ ∣ u ( x ) ∣ p s , μ ∗ ) ∣ u ( x ) ∣ p s , μ ∗ d x ≥ m 0 p ‖ u ‖ V p + b 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ − α ε C p p ‖ u ‖ V p − α C ε C q q ‖ u ‖ V q − β S μ − 2 p s , μ ∗ / p 1 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ .

Taking ε ∈ ( 0 , m 0 / ( 2 p α C p p ) ) , we obtain

J α , β ( u ) ≥ m 0 p ‖ u ‖ V p + ( b − β S μ − 2 p s , μ ∗ / p ) 1 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ − α ε C p p ‖ u ‖ V p − α C ε C q q ‖ u ‖ V q ≥ m 0 2 p + ( b − β S μ − 2 p s , μ ∗ / p ) 1 2 p s , μ ∗ ‖ u ‖ V 2 p s , μ ∗ − p − α C ε C q q ‖ u ‖ V q − p ‖ u ‖ V p .

Since q ∈ ( p , p s ∗ ) and p s ∗ < 2 p s , μ ∗ due to μ ∈ ( 0 , N ) , there exists ϱ > 0 with J α , β ( u ) ≥ ϱ > 0 while u ∈ W V s , p ( R N ) with ‖ u ‖ V = ρ such that 0 < ρ < ‖ e ‖ V = 1 is sufficiently small.

Let

c ≔ inf γ ∈ Γ max t ∈ [ 0 , 1 ] J α , β ( γ ( t ) ) ,

where Γ = { γ ∈ C ( [ 0 , 1 ] , W V s , p ( R N ) ) : γ ( 0 ) = 0 , γ ( 1 ) = e } , i.e., c > 0 . By Lemma 11 and (54), we know that J α , β satisfies the conditions of the mountain pass lemma [39, Theorem 2.1]. So there exists u 2 ∈ W V s , p ( R N ) such that J α , β ( u 2 ) = c > 0 and J α , β ′ ( u 2 ) = 0 . Hence, u 2 is a nontrivial solution of equation (45) with the energy J α , β ( u 2 ) > 0 , which is different from the one with J α , β ( u 1 ) < 0 .□

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  1. Funding information: This work is supported by NSF of China 12071021 and NSF DMS-1820771.

  2. Conflict of interest: The authors declare that there is no conflict of interest in this article.

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Received: 2022-06-08
Revised: 2023-02-27
Accepted: 2023-03-14
Published Online: 2023-04-29

© 2023 the author(s), published by De Gruyter

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

Artikel in diesem Heft

  1. Research Articles
  2. Asymptotic properties of critical points for subcritical Trudinger-Moser functional
  3. The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
  4. On some dense sets in the space of dynamical systems
  5. Sharp profiles for diffusive logistic equation with spatial heterogeneity
  6. Generic properties of the Rabinowitz unbounded continuum
  7. Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
  8. Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
  9. Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
  10. The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
  11. A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
  12. Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
  13. Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
  14. Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
  15. Ground states of Schrödinger systems with the Chern-Simons gauge fields
  16. Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
  17. A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
  18. A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
  19. On a version of hybrid existence result for a system of nonlinear equations
  20. Special Issue: Geometric PDEs and applications
  21. Preface for the special issue on “Geometric Partial Differential Equations and Applications”
  22. Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
  23. Total mean curvatures of Riemannian hypersurfaces
  24. On degenerate case of prescribed curvature measure problems
  25. A curvature flow to the Lp Minkowski-type problem of q-capacity
  26. Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
  27. A note on second derivative estimates for Monge-Ampère-type equations
  28. The Lp chord Minkowski problem
  29. Widths of balls and free boundary minimal submanifolds
  30. Smooth approximation of twisted Kähler-Einstein metrics
  31. The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
  32. A Carleman inequality on product manifolds and applications to rigidity problems
  33. Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
  34. Pinched hypersurfaces are compact
  35. The spinorial energy for asymptotically Euclidean Ricci flow
  36. Geometry of CMC surfaces of finite index
  37. Capillary Schwarz symmetrization in the half-space
  38. Regularity of optimal mapping between hypercubes
  39. Special Issue: In honor of David Jerison
  40. Preface for the special issue in honor of David Jerison
  41. Homogenization of oblique boundary value problems
  42. A proof of a trace formula by Richard Melrose
  43. Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
  44. Regularity properties of monotone measure-preserving maps
  45. Examples of non-Dini domains with large singular sets
  46. Sharp inequalities for coherent states and their optimizers
  47. Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
  48. Propagation of symmetries for Ricci shrinkers
  49. Linear extension operators for Sobolev spaces on radially symmetric binary trees
  50. The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
  51. On an effective equation of the reduced Hartree-Fock theory
  52. Polynomial sequences in discrete nilpotent groups of step 2
  53. Integral inequalities with an extended Poisson kernel and the existence of the extremals
  54. On singular solutions of Lane-Emden equation on the Heisenberg group
https://doi.org/10.1515/ans-2022-0059
Schlagwörter für diesen Artikel
fractional p-Laplacian; multiple solutions; Hardy-Littlewood-Sobolev’s critical exponents; concentration-compactness; genus theory
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Research Articles
  2. Asymptotic properties of critical points for subcritical Trudinger-Moser functional
  3. The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
  4. On some dense sets in the space of dynamical systems
  5. Sharp profiles for diffusive logistic equation with spatial heterogeneity
  6. Generic properties of the Rabinowitz unbounded continuum
  7. Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
  8. Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
  9. Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
  10. The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
  11. A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
  12. Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
  13. Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
  14. Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
  15. Ground states of Schrödinger systems with the Chern-Simons gauge fields
  16. Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
  17. A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
  18. A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
  19. On a version of hybrid existence result for a system of nonlinear equations
  20. Special Issue: Geometric PDEs and applications
  21. Preface for the special issue on “Geometric Partial Differential Equations and Applications”
  22. Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
  23. Total mean curvatures of Riemannian hypersurfaces
  24. On degenerate case of prescribed curvature measure problems
  25. A curvature flow to the Lp Minkowski-type problem of q-capacity
  26. Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
  27. A note on second derivative estimates for Monge-Ampère-type equations
  28. The Lp chord Minkowski problem
  29. Widths of balls and free boundary minimal submanifolds
  30. Smooth approximation of twisted Kähler-Einstein metrics
  31. The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
  32. A Carleman inequality on product manifolds and applications to rigidity problems
  33. Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
  34. Pinched hypersurfaces are compact
  35. The spinorial energy for asymptotically Euclidean Ricci flow
  36. Geometry of CMC surfaces of finite index
  37. Capillary Schwarz symmetrization in the half-space
  38. Regularity of optimal mapping between hypercubes
  39. Special Issue: In honor of David Jerison
  40. Preface for the special issue in honor of David Jerison
  41. Homogenization of oblique boundary value problems
  42. A proof of a trace formula by Richard Melrose
  43. Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
  44. Regularity properties of monotone measure-preserving maps
  45. Examples of non-Dini domains with large singular sets
  46. Sharp inequalities for coherent states and their optimizers
  47. Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
  48. Propagation of symmetries for Ricci shrinkers
  49. Linear extension operators for Sobolev spaces on radially symmetric binary trees
  50. The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
  51. On an effective equation of the reduced Hartree-Fock theory
  52. Polynomial sequences in discrete nilpotent groups of step 2
  53. Integral inequalities with an extended Poisson kernel and the existence of the extremals
  54. On singular solutions of Lane-Emden equation on the Heisenberg group
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