Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells

Yongpeng Chen; Miaomiao Niu

doi:10.1515/math-2022-0006

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Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells

Yongpeng Chen und Miaomiao Niu

Veröffentlicht/Copyright: 7. März 2022

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Abstract

The purpose of this paper is to investigate the ground state solutions for the following nonlinear Schrödinger equations involving the fractional p-Laplacian

( − Δ ) p s u ( x ) + λ V ( x ) u ( x ) p − 1 = u ( x ) q − 1 , u ( x ) ≥ 0 , x ∈ R N ,

where λ > 0 is a parameter, 1 < p < q < N p N − s p , N ≥ 2 , and V ( x ) is a real continuous function on R N . For λ large enough, the existence of ground state solutions are obtained, and they localize near the potential well int ( V − 1 ( 0 ) ) .

Keywords: nonlinear Schrödinger equation; ground state solution; fractional p-Laplacian; variational methods

MSC 2010: 35J60; 35B33

1 Introduction and main results

In this paper, we consider the following nonlinear Schrödinger equations involving the fractional p-Laplacian

(1.1) ( − Δ ) p s u ( x ) + λ V ( x ) u ( x ) p − 1 = u ( x ) q − 1 , x ∈ R N , u ( x ) ≥ 0 , u ( x ) ∈ W s , p ( R N ) ,

where λ > 0 is a parameter, 1 < p < q < N p N − s p , N ≥ 2 , and V ( x ) is a real continuous function on R N .

We are interested in the existence of ground state solutions for λ big enough, and their asymptotical behavior as λ → ∞ . As far as we know, these kinds of problems were first put forward in [1] by Bartsch and Wang, where they studied the Schrödinger equations. Under suitable conditions imposed on the potential, the loss of compactness caused by the whole space R N can be recovered when parameter λ is big enough. Then, many authors began studying the problems with potential well. A lot of results have been obtained.

Bartsch and Parnet [2] also considered the nonlinear Schrödinger equation:

− Δ u + ( a 0 ( x ) + λ a ( x ) ) u = f ( x , u ) , x ∈ R N , u ( x ) → 0 , ∣ x ∣ → ∞ ,

where a 0 ( x ) + λ a ( x ) is indefinite. By using a local linking theorem and the critical groups theory, they obtained the existence of solutions and their asymptotical behavior as λ → ∞ .

Xu and Chen [3] studied the following Kirchhoff problem:

− a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + λ V ( x ) u = f ( x , u ) , x ∈ R N , u ( x ) ∈ H 1 ( R N ) ,

where f ( x , u ) can be sublinear or superlinear. By using the genus theory, they obtained infinitely many negative solutions.

Aleves et al. [4] dealt with the following Choquard equation:

− Δ u + ( λ a ( x ) + 1 ) u = 1 ∣ x ∣ μ ∗ ∣ u ∣ p ∣ u ∣ p − 2 u , x ∈ R N , u ( x ) ∈ H 1 ( R N ) ,

where μ ∈ ( 0 , 3 ) , p ∈ ( 2 , 6 − μ ) , and the potential well Ω = ⋃ j = 1 k Ω j . They proved the existence of a solution, which is nonzero on any subset Ω j . Furthermore, its asymptotical behavior was investigated.

Zhao et al. [5] studied the Schrödinger-Poisson system allowing the potential V ( x ) changes sign

− Δ u + λ V ( x ) u + K ( x ) ϕ u = ∣ u ∣ p − 2 u in R 3 , − Δ ϕ = K ( x ) u 2 in R 3 ,

where p ∈ ( 3 , 6 ) and V ∈ C ( R 3 , R ) are bounded from below. Using the variational method, they obtained the existence and asymptotic behavior of nontrivial solutions.

For the critical problems, Clapp and Ding [6] have studied the nonlinear Schrödinger equation:

− Δ u + λ V ( x ) u = μ u + u 2 ∗ − 1 , x ∈ R N

for N ≥ 4 , λ , μ > 0 . By using variational methods, the authors established existence and multiplicity of positive solutions, which localize near the potential well for λ large and μ small.

Later, the corresponding results obtained in [6] were generalized to the fractional Schrödinger equations by Niu and Tang [7], where they have studied

( − Δ ) s u + ( λ V ( x ) − μ ) u = ∣ u ∣ 2 s ∗ − 2 u , x ∈ R N , u ≥ 0 , u ∈ H s ( R N ) .

Under the linear perturbation, [6] and [7] obtained the existence of solutions and their asymptotic behavior. For the nonlinear perturbation, Alves and Barros [8] considered

− Δ u + λ V ( x ) u = μ u p − 1 + u 2 ∗ − 1 , x ∈ R N .

By employing the Ljusternik-Schnirelmann category, for λ big enough and μ small enough, the aforementioned problem has at last cat ( Ω ) positive solutions.

For more results about these kinds of problems and fractional Schrödinger equations, see, for example, [9,10, 11,12,13, 14,15,16, 17,18,19, 20,21,22, 23,24] and references therein. Motivated by the aforementioned results, we consider equation (1.1). The potential function V ( x ) satisfies

V ( x ) ∈ C ( R N , R ) such that V ( x ) ≥ 0 , Ω ≔ int V − 1 ( 0 ) is a nonempty open set of class C 0.1 with bounded boundary and V − 1 ( 0 ) = Ω ¯ ;
There exists M 0 > 0 such that
μ ( { x ∈ R N : V ( x ) ≤ M 0 } ) < ∞ ,
where μ denotes the Lebesgue measure on R N .

We first introdcue some notations. For s ∈ ( 0 , 1 ) , p ∈ [ 1 , + ∞ ) , define

W s , p ( R N ) ≔ u ∈ L p ( R N ) : ∣ u ( x ) − u ( y ) ∣ ∣ x − y ∣ N p + s ∈ L p ( R N × R N ) ,

endowed with the norm

‖ u ‖ W s , p ( R N ) ≔ ∫ R N ∣ u ∣ p d x + ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + s p d x d y 1 p ,

where the term

[ u ] W s , p ( R N ) ≔ ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y 1 p

is the so-called Gagliardo (semi)norm of u . Moreover, we define

W s , p ( Ω ) ≔ u ∈ L p ( Ω ) : ∣ u ( x ) − u ( y ) ∣ ∣ x − y ∣ N p + s ∈ L p ( Ω × Ω ) ,

endowed with the norm

‖ u ‖ W s , p ( Ω ) ≔ ∫ Ω ∣ u ∣ p d x + ∫ Ω ∫ Ω ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + s p d x d y 1 p .

Let

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

with the norm

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

The energy functional associated with (1.1) is

(1.2) J λ ( u ) = 1 p ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + λ p ∫ R N V ( x ) ∣ u ( x ) ∣ p d x − 1 q ∫ R N u + ( x ) q d x for u ∈ E λ ,

where u + = max { u , 0 } . Then, we can define the Nehari manifold

ℳ λ ≔ u ∈ E λ ⧹ { 0 } : ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + λ ∫ R N V ( x ) ∣ u ( x ) ∣ p d x = ∫ R N u + ( x ) q d x

and

c λ ≔ inf { J λ ( u ) : u ∈ ℳ λ } .

Consider the following “limit” problem of (1.1)

(1.3) ( − Δ ) p s u ( x ) = u ( x ) q − 1 , x ∈ Ω , u ( x ) ≥ 0 , x ∈ Ω , u ( x ) = 0 , x ∈ R N ⧹ Ω ,

Define a subspace E 0 of W s , p ( R N ) as follows:

(1.4) E 0 ≔ { u ∈ W s , p ( R N ) : u ( x ) = 0 in R N ⧹ Ω }

tr Ω E 0 = { u ∣ Ω : u ∈ E 0 } .

The energy functional associated with (1.3) can be defined by

Φ ( u ) = 1 p ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y − 1 q ∫ Ω u + ( x ) q d x for u ∈ E 0 .

Then, the associated Nehari manifold is

N ≔ u ∈ E 0 ⧹ { 0 } : ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y = ∫ Ω u + ( x ) q d x

and

c ( Ω ) ≔ inf { Φ ( u ) : u ∈ N } .

Definition 1.1

A function u λ ( x ) is a ground state solution of (1.1) if c λ is achieved by u λ ∈ ℳ λ , which is a critical point of J λ . Similarly, a function u ( x ) is a ground state solution of (1.3) if c ( Ω ) is achieved by u ∈ N , which is a critical point of Φ .

Definition 1.2

Let X be a Banach space, φ ∈ C 1 ( X , R ) . The function φ satisfies the ( P S ) c condition if any sequence { u n } ⊆ X , such that

(1.5) φ ( u n ) → c , φ ′ ( u n ) → 0

has a convergent subsequence. The sequence { u n } that satisfies (1.5) is called to be a ( P S ) c sequence of φ .

Our main results read as follows:

Theorem 1.3

Suppose ( V 1 ) and ( V 2 ) hold, then for λ large, (1.1) has a ground state solution u λ ( x ) . Furthermore, any sequence λ n → ∞ , { u λ n ( x ) } has a subsequence such that u λ n converges in W s , p ( R N ) along the subsequence to a ground state solution u of (1.3).

Theorem 1.4

Suppose ( V 1 ) and ( V 2 ) hold. Let u n , n ∈ N be a sequence of solutions of (1.1) with λ being replaced by λ n ( λ n → ∞ as n → ∞ ) such that lim sup n → ∞ J λ ( u n ) < ∞ . Then, u n ( x ) converges strongly along a subsequence in W s , p ( R N ) to a solution u of (1.3).

The following paper is organized as follows: In Section 2, we will give some preliminary results. Section 3 is devoted to the “limit” problem, and Section 4 contains the proofs of the main results. C denotes various generic positive constants, and o ( 1 ) will be used to represent quantities that tend to 0 as λ ( or n ) → ∞ .

2 Preliminary results

Lemma 2.1

Let λ 0 > 0 be a fixed constant. Then, for λ ≥ λ 0 > 0 , V ( x ) satisfying ( V 1 ) and ( V 2 ) , E λ is continuously embedded in W s , p ( R N ) uniformly in λ .

Proof

By the definition of W s , p ( R N ) and E λ , we only need to prove the following inequality:

(2.1) ∫ R N ∣ u ( x ) ∣ p d x ≤ C ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + ∫ R N λ V ( x ) ∣ u ( x ) ∣ p d x .

Define

D ≔ { x ∈ R N : V ( x ) ≤ M 0 }

and

D δ 0 ≔ { x ∈ R N : dist ( x , D ) ≤ δ 0 } .

Take ζ ∈ C ∞ ( R N , R ) , 0 ≤ ζ ≤ 1 , satisfying

(2.2) ζ ( x ) = 1 , x ∈ D , 0 , x ∉ D δ 0 , ∣ ∇ ζ ∣ ≤ C / δ 0 .

Then, for any function u ∈ E λ , we can obtain

(2.3) ∫ R N ( 1 − ζ p ) ∣ u ( x ) ∣ p d x = ∫ R N ⧹ D ( 1 − ζ p ) ∣ u ( x ) ∣ p d x + ∫ D ( 1 − ζ p ) ∣ u ( x ) ∣ p d x ≤ 1 λ 0 M 0 λ ∫ R N V ( x ) ∣ u ( x ) ∣ p d x

and

(2.4) ∫ R N ζ p ∣ u ( x ) ∣ p d x = ∫ D δ 0 ζ p ∣ u ( x ) ∣ p d x ≤ μ ( D δ 0 ) 1 − p p s ∗ ∫ D δ 0 ∣ u ( x ) ∣ p s ∗ d x p p s ∗ ≤ C μ ( D δ 0 ) 1 − p p s ∗ ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y ,

where we have used ( V 2 ) and the Sobolev trace inequality

∫ R N ∣ u ( x ) ∣ p s ∗ d x 1 / p s ∗ ≤ C ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y 1 / p ,

for u ∈ W s , p ( R N ) and C = C ( N , p , s ) > 0 . Thus, (2.1) follows from (2.3) and (2.4).□

Lemma 2.2

There exists σ > 0 independent of λ , such that ‖ u ‖ λ ≥ σ for all u ∈ ℳ λ .

Proof

From Lemma 2.1, for any u ∈ ℳ λ ,

0 = ⟨ J λ ′ ( u ) , u ⟩ = ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + λ ∫ R N V ( x ) ∣ u ( x ) ∣ p d x − ∫ R N u + ( x ) q d x ≥ ‖ u ‖ λ p − C ‖ u ‖ W s , p ( R N ) q ≥ ‖ u ‖ λ p − C ‖ u ‖ λ q ,

where C > 0 is independent of λ ≥ 0 . The aforementioned inequality implies that ‖ u ‖ λ q − p ≥ 1 C . Choosing σ = 1 C 1 q − p , we obtain ‖ u ‖ λ ≥ σ .□

Lemma 2.3

Let λ 0 be a fixed positive constant, there exists c 0 > 0 independent of λ ≥ λ 0 > 0 , such that if { u n } is a ( P S ) c sequence of J λ , then either c ≥ c 0 or c = 0 . Moreover,

(2.5) lim sup n → ∞ ‖ u n ‖ λ p ≤ p q q − p c .

Proof

From the definition of ( P S ) c sequence,

c + ‖ u n ‖ λ ⋅ o ( 1 ) = J λ ( u n ) − 1 q ⟨ J λ ′ ( u n ) , u n ⟩ = 1 p − 1 q ∫ R N ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + λ ∫ R N V ( x ) ∣ u n ( x ) ∣ p d x = q − p p q ‖ u n ‖ λ p .

Then, (2.5) holds. On the other side, there is a constant C > 0 independent of λ ≥ λ 0 > 0 , such that

⟨ J λ ′ ( u ) , u ⟩ = ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + λ ∫ R N V ( x ) ∣ u ( x ) ∣ p d x − ∫ R N u + ( x ) q d x ≥ ‖ u ‖ λ p − C ‖ u ‖ λ q .

Thus, there exists σ 1 > 0 independent of λ , such that

(2.6) 1 4 ‖ u ‖ λ p ≤ ⟨ J λ ′ ( u ) , u ⟩ for ‖ u ‖ λ < σ 1 .

If c < σ 1 p ( q − p ) p q , then

lim sup n → ∞ ‖ u n ‖ λ p ≤ c p q q − p < σ 1 p .

Hence, ‖ u n ‖ λ < σ 1 for n large. It follows from (2.6) that

1 4 ‖ u n ‖ λ p ≤ ⟨ J λ ′ ( u n ) , u n ⟩ = o ( 1 ) ‖ u n ‖ λ ,

which implies ‖ u n ‖ λ → 0 as n → ∞ . Therefore, J λ ( u n ) → 0 , that is, c = 0 . Thus, c 0 = σ 1 p ( q − p ) q p is as required.□

Lemma 2.4

There exists δ 0 > 0 , such that any ( P S ) c sequence { u n } of J λ with λ ≥ 0 and c > 0 satisfies

(2.7) lim inf n → ∞ ‖ u n + ‖ L q ( R N ) q ≥ δ 0 c .

Proof

From the definition of ( P S ) c sequence,

c = lim n → ∞ J λ ( u n ) − 1 p ⟨ J λ ′ ( u n ) , u n ⟩ = 1 p − 1 q lim n → ∞ ∫ R N u n + ( x ) q d x = ( q − p ) q p lim n → ∞ ‖ u n + ( x ) ‖ L q ( R N ) q ,

which implies (2.7) with δ 0 ≤ q p q − p .□

Lemma 2.5

Let C 1 be any fixed constant. Then, for any ε > 0 , there exists Λ ε > 0 and R ε > 0 , such that if { u n } is a ( P S ) c sequence of J λ with λ ≥ Λ ε , c ≤ C 1 , then

(2.8) lim sup n → ∞ ∫ B R ε c u n + ( x ) q d x ≤ ε ,

where B R ε c = { x ∈ R N : ∣ x ∣ ≥ R ε } .

Proof

For R > 0 , let

A ( R ) ≔ { x ∈ R N : ∣ x ∣ > R , V ( x ) ≥ M 0 }

and

B ( R ) ≔ { x ∈ R N : ∣ x ∣ > R , V ( x ) < M 0 } .

It follows from Lemma 2.3 that

(2.9) ∫ A ( R ) ∣ u n ( x ) ∣ p d x ≤ 1 λ M 0 ∫ R N λ V ( x ) ∣ u n ( x ) ∣ p d x ≤ 1 λ M 0 ∫ R N ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + ∫ R N λ V ( x ) ∣ u n ( x ) ∣ p d x ≤ 1 λ M 0 p q q − p C 1 + o ( 1 ) .

From Hölder inequality and (2.5), we can see that, for 1 < r < N / ( N − p s ) ,

(2.10) ∫ B ( R ) ∣ u n ( x ) ∣ p d x ≤ ∫ R N ∣ u n ( x ) ∣ p r d x 1 / r μ ( B ( R ) ) 1 / r ′ ≤ C ‖ u n ‖ λ p ⋅ μ ( B ( R ) ) 1 / r ′ ≤ C p q q − p C 0 ⋅ μ ( B ( R ) ) 1 / r ′ ,

where C = C ( N , r ) > 0 and 1 / r + 1 / r ′ = 1 . By interpolation inequality and Sobolev embedding inequality, we can obtain

∫ B R c u n + ( x ) q d x ≤ ∫ B R c ∣ u n ( x ) ∣ p d x q ( 1 − θ ) p ⋅ ∫ B R c ∣ u n ( x ) ∣ p s ∗ d x q θ p s ∗ ≤ ∫ B R c ∣ u n ( x ) ∣ p d x q ( 1 − θ ) p ∫ R N ∣ u n ( x ) ∣ p s ∗ d x q θ p s ∗ ≤ C ∫ B R c ∣ u n ( x ) ∣ p d x q ( 1 − θ ) p ∫ R N ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y q θ p ≤ C ∫ A ( R ) ∣ u n ( x ) ∣ p d x + ∫ B ( R ) ∣ u n ( x ) ∣ p d x q ( 1 − θ ) p ‖ u n ‖ λ q θ ,

where θ = N s q − p p q . Then, the result follows from (2.9), (2.10) and ( V 2 ) .□

Lemma 2.6

(Brézis-Lieb lemma, 1983) Let { u n } ⊂ L p ( R N ) , 1 ≤ p < ∞ . If

{ u n } is bounded in L p ( R N ) ,
u n → u almost everywhere on R N , then
(2.11) lim n → ∞ ( ∣ u n ∣ p p − ∣ u n − u ∣ p p ) = ∣ u ∣ p p .

Lemma 2.7

Let λ ≥ λ 0 > 0 be fixed and let { u n } be a ( P S ) c sequence of J λ . Then, up to a subsequence, u n ⇀ u in E λ with u being a weak solution of (1.1). Moreover, u n 1 = u n − u is ( P S ) c ′ sequence with c ′ = c − J λ ( u ) .

Proof

By Lemma 2.3, { u n } is bounded in E λ . Then, up to a subsequence u n ⇀ u in E λ as n → ∞ , and

(2.12) u n ⇀ u in W s , p ( R N ) ,

(2.13) u n ⇀ u in L q ( R N ) , p ≤ q < p s ∗ ,

(2.14) u n → u in L loc q ( R N ) , p ≤ q < p s ∗ ,

(2.15) u n → u a.e. in R N ,

where p s ∗ = N p N − p s is the fractional critical Sobolev exponent. Hence, for any φ ∈ E λ , we have

⟨ J λ ′ ( u n ) , φ ⟩ = ∫ R N ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + p s d x d y + λ ∫ R N V ( x ) ∣ u n ( x ) ∣ p − 2 u n ( x ) φ ( x ) d x − ∫ R N u n + ( x ) q − 1 φ ( x ) d x → ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + p s d x d y + λ ∫ R N V ( x ) u ( x ) p − 1 φ ( x ) d x − ∫ R N u + ( x ) q − 1 φ ( x ) d x = ⟨ J λ ′ ( u ) , φ ⟩ .

Therefore,

(2.16) ⟨ J λ ′ ( u ) , φ ⟩ = lim n → ∞ ⟨ J λ ′ ( u n ) , φ ⟩ = 0 ,

which implies that u is a critical point of J λ .

Let u n 1 = u n − u , we will show that as n → ∞ ,

(2.17) J λ ( u n 1 ) → c − J λ ( u )

and

(2.18) J λ ′ ( u n 1 ) → 0 .

To show (2.17), we observe that

(2.19) J λ ( u n 1 ) = 1 p ∫ R N ∫ R N ∣ u n 1 ( x ) − u n 1 ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + λ p ∫ R N V ( x ) ∣ u n 1 ( x ) ∣ p d x − 1 q ∫ R N u n 1 + ( x ) q d x = J λ ( u n ) − J λ ( u ) + λ p ∫ R N V ( x ) ( ∣ u n 1 ( x ) ∣ p − ∣ u n ( x ) ∣ p + ∣ u ( x ) ∣ p ) d x + 1 p ∫ R N ∫ R N ∣ u n 1 ( x ) − u n 1 ( y ) ∣ p − ∣ u n ( x ) − u n ( y ) ∣ p + ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + 1 q ∫ R N u n + ( x ) q d x − 1 q ∫ R N u n 1 + ( x ) q d x − 1 q ∫ R N u + ( x ) q d x .

From Lemma 2.6, ∫ R N u n + ( x ) q d x − ∫ R N u + ( x ) q d x − ∫ R N u n 1 + ( x ) q d x → 0 as n → ∞ . Conversely, we know that ‖ u n ‖ λ p − ‖ u ‖ λ p − ‖ u n 1 ‖ λ p → 0 , as n → ∞ . Thus, from (2.19), we indeed have obtained (2.17). Now we come to show (2.18). From (2.16), we have for any φ ∈ E λ

⟨ J λ ′ ( u n 1 ) , φ ⟩ = ⟨ J λ ′ ( u n ) , φ ⟩ − ∫ R N ( u n 1 + ) q − 1 φ ( x ) d x + ∫ R N ( u n + ) q − 1 φ ( x ) d x − ∫ R N ( u + ) q − 1 φ ( x ) d x + o ( 1 ) .

Since J λ ′ ( u n ) → 0 and u n ⇀ u in L q ( R N ) , we have

lim n → ∞ sup ‖ φ ‖ λ ≤ 1 ∫ R N ( ( u n 1 + ) q − 1 ( x ) φ ( x ) − ( u n + ) q − 1 φ ( x ) + ( u + ) q − 1 φ ( x ) ) d x = 0 .

Thus, we have

lim n → ∞ ⟨ J λ ′ ( u n 1 ) , φ ⟩ = 0 for any φ ∈ E λ ,

which implies (2.18), and this completes the proof.□

Proposition 2.8

Suppose ( V 1 ) and ( V 2 ) hold. Then, for any C 0 > 0 , there exists Λ 0 > 0 such that J λ satisfies the ( P S ) c condition for all λ ≥ Λ 0 and c ≤ C 0 .

Proof

Choose 0 < ε < δ 0 c 0 / 2 , where c 0 and δ 0 are the constants in Lemmas 2.3 and 2.4, respectively. Let Λ 0 ≔ Λ ε , where Λ ε > 0 is from Lemma 2.5.

Assume { u n } is a ( P S ) c sequence of J λ with λ ≥ Λ 0 and c ≤ C 0 . By Lemma 2.7, u n 1 = u n − u is a ( P S ) c ′ sequence of J λ with c ′ = c − J λ ( u ) . If c ′ > 0 , it follows from Lemma 2.3 that c ′ ≥ c 0 . From Lemma 2.4, we can obtain

lim inf n → ∞ ‖ u n 1 + ( ⋅ ) ‖ L q ( R N ) q ≥ δ 0 c ′ ≥ δ 0 c 0 .

Conversely, Lemma 2.5 implies

lim sup n → ∞ ∫ B R ε c u n 1 + ( x ) q ≤ ε < δ 0 c 0 2 .

Noting u n 1 → 0 in L loc q ( R N ) , p ≤ q < p s ∗ , a contradiction follows from the aforementioned two inequalities. Therefore, c ′ = 0 . Thus, u n 1 → 0 in E λ by Lemma 2.3.□

Corollary 2.9

For any q ∈ ( p , p s ∗ ) , there exists Λ 0 > 0 , such that c λ is achieved for all λ ≥ Λ 0 at some u λ ∈ E λ , which is a ground state solution of (1.1).

Proof

By Ekeland variational principle, there is a PS sequence u n ∈ E λ , such that

J λ ( u n ) → c λ and J λ ′ ( u n ) → 0 .

By Proposition 2.8, there exists some u λ ∈ E λ , such that, up to subsequence, u n → u λ in E λ as n → ∞ and λ is sufficiently large. It is not difficult to show that

J λ ( u n ) → J λ ( u λ ) and J λ ′ ( u n ) → J λ ′ ( u λ ) .

Therefore, we have J λ ( u λ ) = c λ and J λ ′ ( u λ ) = 0 . This means that u λ is a ground state solution of (1.1).□

3 Limit problem

Lemma 3.1

Let 1 < p < q < p s ∗ ≔ p N N − p s , N ≥ 2 . Then, t r Ω E 0 is compactly embedded in L q ( Ω ) .

Proof

Since t r Ω E 0 ⊂ W s , p ( Ω ) and W s , p ( Ω ) ↪ L p ( Ω ) are compact for p < q < p s ∗ , N ≥ 2 , the result follows.□

Lemma 3.2

The infimum c ( Ω ) is achieved by a function u ∈ N , which is a ground state solution of (1.3).

Proof

By Ekeland variational principle, there is a PS sequence u n ∈ E 0 , such that

Φ ( u n ) → c ( Ω ) and Φ ′ ( u n ) → 0 .

Thus, by Lemma 3.1, we can easily obtain a subsequence of { u n } (still denote it itself), such that u n → u in E 0 . Therefore, u is a ground state solution of (1.3).□

Remark 3.3

Assume set Ω = int V − 1 ( 0 ) has more than one isolated component, for example, Ω = Ω 1 ∪ Ω 2 with Ω 1 ∩ Ω 2 = ∅ . Suppose that u ∈ N is a nonnegative solution of (1.3) with u ( x ) = 0 in Ω 1 and u ( x ) ≩ 0 in Ω 2 . Then, we have ( − Δ ) p s u ( x ) = ∫ R N ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ∣ x − y ∣ N + p s d y < 0 in Ω 1 . Conversely, ( − Δ ) p s u ( x ) = u ( x ) q − 1 = 0 for x ∈ Ω 1 . This contradiction shows that the nonnegative solution u ( x ) of (1.3) must be u ( x ) ≩ 0 in both Ω 1 and Ω 2 . However, the Laplacian case can have a nonnegative solution u ( x ) satisfying u ( x ) = 0 in Ω 1 and u ( x ) ≩ 0 in Ω 2 . The difference between the two phenomena is attributed to the nonlocality of fractional operators and the locality of Laplacian operators.

4 The proof of the main results

Lemma 4.1

c λ → c ( Ω ) as λ → ∞ .

Proof

From the definition of c λ and c ( Ω ) , we know that c λ ≤ c ( Ω ) , λ > 0 . Furthermore, c λ is monotone increasing about the parameter λ > 0 . Then, there exists a constant k , such that

lim n → ∞ c λ n = k ,

where λ n → ∞ . It follows from Lemma 2.3 that k > 0 . By Corollary 2.9, for n large enough, there exists a sequence u n ∈ ℳ λ n , such that J λ n ′ ( u n ) = 0 and J λ n ( u n ) = c λ n . If k < c ( Ω ) , it is easy to see that { u n } is bounded in W s , p ( R N ) ; thus, we can assume that u n ⇀ u in E and

(4.1) u n ( x ) → u ( x ) in L loc θ ( R N ) for p ≤ θ < p s ∗ .

Claim 1: u ∣ Ω c = 0 . In fact, if u ∣ Ω c ≠ 0 , then there exists a compact subset F ⊂ Ω c with dist ( F , Ω ) > 0 , such that u ∣ F ≠ 0 . It follows from (4.1) that

∫ F ∣ u n ( x ) ∣ p d x → ∫ F ∣ u ( x ) ∣ p d x > 0 .

However, there exists ε 0 > 0 , such that V ( x ) ≥ ε 0 > 0 , x ∈ F . Thus,

J λ n ( u n ) ≥ q − p p q λ n ∫ F V ( x ) ∣ u n ( x ) ∣ p d x ≥ q − p p q λ n ε 0 ∫ F ∣ u n ( x ) ∣ p d x → ∞ as n → ∞ ,

which is a contradiction. Therefore, u ∈ E 0 .

Claim 2: u n → u in L q ( R N ) for p < q < p s ∗ . Indeed, if not, then by the concentration-compactness lemma from the study by Loins [25], there exist δ > 0 , ρ > 0 and x n ∈ R N with ∣ x n ∣ → ∞ , such that

lim inf n → ∞ ∫ B ρ ( x n ) ∣ u n ( x ) − u ( x ) ∣ p d x ≥ δ > 0 .

Then, we have

J λ n ( u n ) = q − p p q ∫ R N ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + q − p p q ∫ R N λ n V ( x ) ∣ u n ( x ) ∣ p d x ≥ q − p p q λ n ∫ B ρ ( x n ) ∩ { x : V ( x ) ≥ M 0 } V ( x ) ∣ u n ( x ) ∣ p d x = q − p p q λ n ∫ B ρ ( x n ) ∩ { x : V ( x ) ≥ M 0 } V ( x ) ∣ u n ( x ) − u ( x ) ∣ p d x ≥ q − p p q λ n M 0 ∫ B ρ ( x n ) ∣ u n ( x ) − u ( x ) ∣ p d x − M 0 ∫ B ρ ( x n ) ∩ { x : V ( x ) ≤ M 0 } ∣ u n ( x ) ∣ p d x ≥ q − p p q λ n M 0 ∫ B ρ ( x n ) ∣ u n ( x ) − u ( x ) ∣ p d x − o ( 1 ) → ∞ as n → ∞ ,

as a contradiction. So u n → u in L p ( R N ) . Therefore, it is easy to see that u ≥ 0 is a solution for problem (1.3). Furthermore,

k = lim n → ∞ c λ n = lim n → ∞ J λ n ( u n ) = lim n → ∞ 1 p − 1 q ∫ R N u n + ( x ) q d x = 1 p − 1 q ∫ Ω u + ( x ) q d x ,

which means u ∈ N , and then, k ≥ c ( Ω ) , a contradiction. Hence, lim λ → ∞ c λ = c ( Ω ) .□

Proof of Theorem 1.3

By Corollary 2.9, there exists u n ∈ ℳ λ n , such that J λ n ( u n ) = c λ n ( λ n → ∞ as n → ∞ ). It is easy to see that { u n } is bounded in W s , p ( R N ) . Then, without loss of generality, u n ⇀ u in W s , p ( R N ) and u n → u in L loc θ ( R N ) for p < θ < p s ∗ .

Now we prove that u n → u strongly in W s , p ( R N ) and u is a ground state solution of (1.3). First, as the proof of Lemma 4.1, u ≥ 0 is a solution of problem (1.3) and u n + → u + strongly in L q ( R N ) .

Now we claim that

λ n ∫ R N V ( x ) ∣ u n ( x ) ∣ p d x → 0

and

∫ R N ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y → ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y .

Indeed, if either

lim inf n → ∞ λ n ∫ R N V ( x ) ∣ u n ( x ) ∣ p d x > 0

lim inf n → ∞ ∫ R N ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y > ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y .

Thus, we have

∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y < ∫ Ω u + ( x ) q d x .

Therefore, there is α ∈ ( 0 , 1 ) , such that α u ∈ N and

c ( Ω ) ≤ Φ ( α u ) = q − p p q ∫ R N ∫ R N ∣ α u ( x ) − α u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y < q − p p q ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ n + p s d x d y ≤ lim n → ∞ q − p p q ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ N + p s d x d y + ∫ R N λ n V ( x ) ∣ u n ( x ) ∣ p d x = lim n → ∞ J λ n ( u n ) = c ( Ω ) ,

which is a contradiction. By now we complete the proof of Theorem 1.3.□

Proof of Theorem 1.4

Suppose { u n } ⊂ W s , p ( R N ) is a solution of (1.1) with λ being replaced by λ n ( λ n → ∞ as n → ∞ ). It follows from lim sup n → ∞ J λ ( u n ) < ∞ that such a sequence { u n } is bounded in W s , p ( R N ) . Suppose that u n ⇀ u in W s , p ( R N ) and u n → u in L loc θ ( R N ) for p < θ < p s ∗ . Similar to the proof of Lemma 4.1, u ∣ Ω c = 0 and u ∈ E 0 is solution of (1.3). Moreover, u n → u in L θ ( R N ) for p < θ < p s ∗ . Noting u n ∈ ℳ λ n and u ∈ N , we can obtain

∫ R N ∫ R N ∣ u n ( x ) − u n ( y ) − u ( x ) + u ( y ) ∣ p ∣ x − y ∣ n + p s d x d y + ∫ R N λ n V ( x ) ∣ u n ( x ) − u ( x ) ∣ p d x = ∫ R N ∫ R N ∣ u n ( x ) − u n ( y ) ∣ p ∣ x − y ∣ n + p s d x d y + ∫ R N λ n V ( x ) ∣ u n ( x ) ∣ p d x − ∫ R N ∫ R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ n + p s d x d y − ∫ R N λ n V ( x ) ∣ u ( x ) ∣ p d x + o ( 1 ) = ∫ R N u n + ( x ) q d x − ∫ Ω u + ( x ) q d x + o ( 1 ) = o ( 1 ) .

Thus, u n → u in W s , p ( R N ) . This completes the proof of Theorem 1.4.□

Acknowledgments

The authors would like to thank the anonymous referee for his/her careful readings of our manuscript and the useful comments made for its improvement.

Funding information: Y. Chen was supported by National Natural Science Foundation of China (12161007), Basic Ability Improvement Project of Young and Middle-aged Teachers in Guangxi Universities (2021KY0348), Doctoral Foundation of Guangxi University of Science and Technology (20Z30), and Guangxi Science and Technology Base and Talent Project (AD21238019). M. Niu was supported by National Natural Science Foundation of China (11901020), Beijing Natural Science Foundation (1204026), and the Science and Technology Project of Beijing Municipal Commission of Education China (No. KM202010005027).
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-07-10

Revised: 2021-11-28

Accepted: 2021-12-14

Published Online: 2022-03-07

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https://doi.org/10.1515/math-2022-0006

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nonlinear Schrödinger equation; ground state solution; fractional p-Laplacian; variational methods

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