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Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system

  • Jian Zhang EMAIL logo , Huitao Zhou and Heilong Mi
Published/Copyright: March 12, 2024
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Advances in Nonlinear Analysis
From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

This article is concerned with the following Hamiltonian elliptic system:

− ε 2 Δ u + ε b → ⋅ ∇ u + u + V ( x ) v = H v ( u , v ) in R N , − ε 2 Δ v − ε b → ⋅ ∇ v + v + V ( x ) u = H u ( u , v ) in R N ,

where ε > 0 is a small parameter, V is a potential function, and H is a super-quadratic sub-critical Hamiltonian. Applying suitable variational arguments and refined analysis techniques, we construct a new multiplicity result of semiclassical solutions which depends on the number of global minimum points of V . This result indicates how the shape of the graph of V affects the number of semiclassical solutions.

Keywords: Hamiltonian elliptic system; semiclassical solutions; multiplicity
MSC 2010: 35J50; 35Q40; 58E05

1 Introduction and main result

In this article, we deal with a class of singularly perturbed Hamiltonian elliptic system with gradient term

(1.1) − ε 2 Δ u + ε b → ⋅ ∇ u + u + V ( x ) v = H v ( u , v ) in R N , − ε 2 Δ v − ε b → ⋅ ∇ v + v + V ( x ) u = H u ( u , v ) in R N ,

where z = ( u , v ) : R N → R 2 , ε > 0 is a small positive parameter, b → is a constant vector, V is a potential function, and H u and H v denote the partial derivatives of H with respect to u and v . The main motivation for the study of model (1.1) is that its solutions are in fact the static states for the following reaction–diffusion system:

∂ t u − Δ x u + b → ( t , x ) ⋅ ∇ u + V ( x ) u = H v ( t , x , u , v ) in R × R N , − ∂ t v − Δ x v − b → ( t , x ) ⋅ ∇ v + V ( x ) v = H u ( t , x , u , u ) in R × R N ,

which is applied to model chemical concentrations due to reaction and diffusion. The function V is considered as the chemical potential, and H represents the external physicochemical interaction. Moreover, it also appears in various fields, such as physics and chemistry, quantum mechanics, control theory, and Brownian motions. For more detailed contents and applications in the physical science and other fields, we refer the readers to see the monographs of Nagasawa [19] and Lions [18].

In the past few decades, Hamiltonian elliptic systems like (1.1) have attracted considerable interest due to many powerful applications in different fields, the literature studies related to these systems are enormous and encompass several interesting topics of study in nonlinear analysis, including existence, nonexistence, multiplicity, and finer qualitative properties of solutions.

When ε = 1 , De Figueiredo and Felmer [10] and Hulshof and De Vorst [13] studied the elliptic system defined on the bounded domain and obtained the existence result of nontrivial solutions by using generalized mountain pass theorem in [5]. Later, the result of multiple solutions was established by De Figueiredo and Ding [9]. For the case that the system is settled on the whole space, we need to deal with the main difficulty caused by the lack of the compactness of the Sobolev embedding. Besides, the main unusual feature of the Hamiltonian system is that the corresponding energy functional is strongly indefinite. Based on the above two features, the standard variational methods like Nehari manifold method and mountain pass theorem are unavailable. Some refined variational arguments were subsequently developed by many scholars for strongly indefinite functionals. We refer to the dual variational method [3,24], the Orlitz space approach [8], the generalized linking theory [4,14], the reduction method [7], and so on. Recently, applying some approaches introduced above, the articles [15–17,21,27,29–32,36] investigated the existence and multiplicity results of nontrivial solutions of system (1.1) under various conditions. For more results we also mention the recent overview by Bonheure et al. [6] for a very comprehensive introduction about Hamiltonian elliptic system.

When ε is small, the standing wave solutions of system (1.1) are called as semiclassical states. One of the basic principles of quantum mechanics is the correspondence principle, which indicates that the laws of quantum mechanics reduce to those of classical mechanics as ε → 0 . The concentration phenomenon of semiclassical states, as ε goes to zero, reflects the transition from quantum mechanics to classical mechanics, which is meaningful in physics and gives rise to important physical insights.

Concerning the investigation of semiclassical solutions for system (1.1) we would like to mention the related works [3,12,22,23,28,35,37,38]. More precisely, Ávila-Yang [3] obtained the existence and boundary concentration behavior of positive solutions for a elliptic system with zero Neumann boundary condition. By means of infinite dimensional Lyapunov-Schmidt reduction method, Ramos and Tavares [22] and Ramos and Soares [23] established the existence of positive solutions which concentrate at local and global minimum points of the potential V . A new concentration pattern that semiclassical solutions concentrate around the local saddle points or local maximum points of the potential V can be found in Zhang and Zhang [38].

Very recently, Zhang et al. [35] showed the existence and concentration (around the maximum points of the nonlinear potential) of solution for the following system with nonlinear potential. Further results to system with competing potentials (including linear potential and nonlinear potential) have also appeared in [37], in which the semiclassical ground state solutions concentrating around the global minimum points of linear potential and the global maxima points of nonlinear potential were established under the global condition of the linear potential V

inf x ∈ R N V ( x ) < liminf ∣ x ∣ → ∞ V ( x ) .

Later on, Zhang et al. [28] constructed a family of semiclassical solutions and showed that the concentration phenomena hold around local minimum of V under the local condition of the potential V

min x ∈ Ω V ( x ) < min x ∈ ∂ Ω V ( x ) ,

where Ω is a bounded domain in R N . For other results related to the Hamiltonian elliptic system, we refer to [3,12,33] and references therein

We would like to emphasize that all the works mentioned above only focus on the existence and concentration of semiclassical solutions, but the multiplicity result of semiclassical solutions has not been studied to system (1.1) up to now.

Motivated by the works [28] and [37], our main purpose of this article is to complement the results found in [28] and [37] in the following sense: we intend to establish a new multiplicity result of semiclassical solutions for system (1.1). To be more precise, in the present article we shall study that the number of global minimum points of V is directly related to the number of semiclassical solutions when ε is small, then the multiplicity of solutions can be obtained.

Before stating our result we assume the following conditions hold for the potential V and the nonlinearity H .

  1. V : R N → R is a continuous function such that

    max x ∈ R N ∣ V ( x ) ∣ = V max < 1 and min x ∈ R N V ( x ) = V 0 < V ∞ = lim ∣ x ∣ → ∞ V ( x ) ;

  2. there exist k points x 1 , x 2 , … , x k ∈ R N with x 1 = 0 such that V ( x i ) = V 0 for all 1 ⩽ i ⩽ k .

  3. there exists h ∈ C ( R + , R + ) such that H z ( z ) = h ( ∣ z ∣ ) z , and for some c 0 > 0 and p ∈ ( 2 , 2 * ) , there holds

    ∣ h ( s ) ∣ ⩽ c 0 ( 1 + ∣ s ∣ p − 2 ) for all s ∈ R + ,

    where ( H u ( u , v ) , H v ( u , v ) ) = H z ( z ) and 2 * is usual Sobolev critical exponent;

  4. h ( s ) = o ( 1 ) as s → 0 , and H ( s ) ∕ s 2 → + ∞ as s → + ∞ ;

  5. h ( s ) is strictly increasing in s on ( 0 , + ∞ ) .

We state in what follows the main result of this article.

Theorem 1.1

Assume that conditions ( V 1 ) – ( V 2 ) and ( h 1 ) – ( h 3 ) hold. Then there exists ε 0 > 0 such that system (1.1) has at least k semiclassical solutions for each ε ∈ ( 0 , ε 0 ) .

We would like to point that the result included in this article indicates that how the shape of the graph of V affects the number of semiclassical solutions, and the conclusion of multiplicity of semiclassical solutions complements several recent contributions to the study of Hamiltonian elliptic system with strongly indefinite variational structure.

Next we sketch the strategies and methods to prove the main result. The proof of Theorem 1.1 will be carried out by using suitable variational methods and refined analysis techniques. As described in the previous introduction, the strongly indefinite structure of energy functional and the lack of compactness are two major difficulties we encounter to seek for the existence of semiclassical solutions.

First, we will take advantage of the method of generalized Nehari manifold developed by Szulkin and Weth [25] to conquer the difficulty caused by strongly indefinite feature. It is worth pointing out that some estimates proved in the present article were also inspired by arguments found in [25]; however, it is necessary to be more refined because the problems are different and some estimates cannot be done by the same way as in the article [25]. Second, we have to prove that the energy functional possesses necessary compactness property at some minimax level to resolve the difficulty aroused by the lack of compactness. This key point will be achieved by employing the energy comparison argument to establish some exact comparison relationships of the ground state energy level between the original problem and certain auxiliary problems. Finally, in order to prove the multiplicity result, we obtain several very useful conclusions by using the nice property of barycenter map, which contribute to construct some different Palais-Smale sequences. Furthermore, combining the Ekeland’s variational principle, limit problem’s technique and refined analysis tools, we can construct k semiclassical solutions.

The organization of the remainder of this article is as follows. In Section 2, we present a suitable variational framework associated with system (1.1) and prove some useful preliminary results. In Section 3, we introduce the existence and some properties of the ground state solutions for the constant coefficient system. Section 4 is devoted to the completed proofs of Theorem 1.1.

2 Variational framework and preliminary results

Throughout the present article, we use the following notations which will be used later.

  • ‖ ⋅ ‖ s denotes the usual norm of the Lebesgue space L s ( R N ) for 1 ⩽ s ⩽ + ∞ ;

  • ( ⋅ , ⋅ ) 2 denotes the inner product of L 2 ( R N ) ;

  • c , c i , C i denote (possibly different) any positive constants, whose values are relevant;

  • σ ( A ) and σ e ( A ) denote the spectrum and the essential spectrum of operator A .

In this section, we will introduce the function space which will work for system (1.1) and some preliminary results that are crucial in our approach.

In order to prove the main result, we do not deal with system (1.1) directly, but instead we study an equivalent system with system (1.1). Indeed, using the change of variable x ↦ ε x , we can rewrite system (1.1) as the following equivalent system:

(2.1) − Δ u + b → ⋅ ∇ u + u + V ( ε x ) v = H v ( u , v ) in R N , − Δ v − b → ⋅ ∇ v + v + V ( ε x ) u = H u ( u , v ) in R N .

Evidently, we can see that if z = ( u , v ) is a solution of system (2.1), then

z ^ ( x ) = ( u ^ ( x ) , v ^ ( x ) ) = ( u ( x ∕ ε ) , v ( x ∕ ε ) )

is a solution of system (1.1). Therefore, next we will study the equivalent system (2.1).

To continue the discussion, we introduce the following notations. Let

J = 0 − 1 1 0 and J 0 = 0 1 1 0 ,

and S = − Δ + 1 . We denote

A ≔ S J 0 + b → ⋅ ∇ J = 0 − Δ − b → ⋅ ∇ + 1 − Δ + b → ⋅ ∇ + 1 0 .

Then system (2.1) can be rewritten as

(2.2) A z + V ( ε x ) z = H z ( z ) .

Now we establish the variational framework of system (2.1), we collect some properties of the spectrum of the operator A , whose proofs can be found in [29], so we omit the details.

Lemma 2.1

The operator A is a self-adjoint operator on L 2 ≔ L 2 ( R N , R 2 ) with domain D ( A ) ≔ H 2 ( R N , R 2 ) .

Lemma 2.2

We have the following two conclusions about the spectrum of A :

  1. σ ( A ) = σ e ( A ) , i.e., A has only essential spectrum;

  2. σ ( A ) ⊂ R \ ( − 1 , 1 ) and σ ( A ) is symmetric with respect to origin.

Evidently, it follows from Lemmas 2.1 and 2.2 that the space L 2 possesses the following orthogonal decomposition:

L 2 = L − ⊕ L +

such that A is negative definite (resp. positive definite) in L − (resp. L + ).

Denoting by ∣ A ∣ the absolute value of A and by ∣ A ∣ 1 ∕ 2 its square root of ∣ A ∣ , and let E = D ( ∣ A ∣ 1 ∕ 2 ) be the domain of the self-adjoint operator ∣ A ∣ 1 ∕ 2 , which is a Hilbert space equipped with the inner product

( z , w ) = ( ∣ A ∣ 1 ∕ 2 z , ∣ A ∣ 1 ∕ 2 w ) 2

and norm ‖ z ‖ 2 = ( z , z ) . Since E ⊂ L 2 , it follows that E has the following decomposition:

E = E − ⊕ E + , where E ± = E ∩ L ± ,

which is orthogonal with respect to the inner products ( ⋅ , ⋅ ) 2 and ( ⋅ , ⋅ ) . Moreover, using the polar decomposition of A we can obtain that

A u − = − ∣ A ∣ z − , A u + = ∣ A ∣ z + for all z = z + + z − ∈ E .

Clearly, since σ ( A ) ⊂ R \ ( − 1 , 1 ) , we can know that

(2.3) ‖ z ‖ 2 2 ⩽ ‖ z ‖ 2 for any z ∈ E .

From [29, Lemma 2.4], it follows that ‖ ⋅ ‖ and ‖ ⋅ ‖ H 1 are two equivalent norms. Hence, we have the embedding theorem, that is, E embeds continuously into L p for each p ∈ [ 2 , 2 * ] and compactly into L loc p for each p ∈ [ 1 , 2 * ) . Then, there exists positive constant π p such that

(2.4) ‖ z ‖ p ⩽ π p ‖ z ‖ , for all z ∈ E , p ∈ [ 2 , 2 * ] .

According to conditions ( h 1 ) and ( h 2 ) , we can infer that for any ε > 0 , there is C ε > 0 such that

(2.5) ∣ h ( s ) ∣ ⩽ ε + C ε ∣ s ∣ p − 2 and ∣ H ( s ) ∣ ⩽ ε ∣ s ∣ 2 + C ε ∣ s ∣ p for each s ∈ R + ,

where p ∈ ( 2 , 2 * ) . Moreover, from ( h 3 ) we can deduce that

(2.6) H ( s ) > 0 and 1 2 h ( s ) s 2 − H ( s ) > 0 , ∀ s > 0 .

Under the comments above, we can define the energy functional associated with system (2.1) by

I ε ( z ) = 1 2 ∫ R N A z ⋅ z + 1 2 ∫ R N V ( ε x ) ∣ z ∣ 2 d x − ∫ R N H ( ∣ z ∣ ) d x ,

where z ∈ E and ⋅ denotes the usual inner product in R 2 . Employing the polar decomposition of A , the energy functional I ε has another representation as follows:

I ε ( z ) = 1 2 ( ‖ z + ‖ 2 − ‖ z − ‖ 2 ) + 1 2 ∫ R N V ( ε x ) ∣ z ∣ 2 d x − ∫ R N H ( ∣ z ∣ ) d x .

From Lemma 2.2 we can see that I ε is strongly indefinite. Our hypotheses imply that I ε ∈ C 1 ( E , R ) . By standard argument we know that critical points of I ε are solutions of system (2.1), and for z , ψ ∈ E , there holds

⟨ I ε ′ ( z ) , ψ ⟩ = ( z + , ψ + ) − ( z − , ψ − ) + ∫ R N V ( ε x ) z ψ d x − ∫ R N h ( ∣ z ∣ ) z ψ d x .

We note that if z ∈ E is a nontrivial critical point of I ε , then we can obtain from (2.6) that

I ε ( z ) = I ε ( z ) − 1 2 ⟨ I ε ′ ( z ) , z ⟩ = ∫ R N 1 2 h ( ∣ z ∣ ) ∣ z ∣ 2 − H ( ∣ z ∣ ) d x > 0 .

On the other hand, for any z ∈ E − , using ( V 1 ) , (2.3), and (2.6) we obtain

I ε ( z ) = − 1 2 ‖ z ‖ 2 + 1 2 ∫ R N V ( ε x ) ∣ z ∣ 2 d x − ∫ R N H ( ∣ z ∣ ) d x ⩽ − 1 − V max 2 ‖ z ‖ 2 − ∫ R N H ( ∣ z ∣ ) d x ⩽ 0 .

Therefore, we can see that all nontrivial critical points of I ε are in the space E \ E − .

Next we consider the following set introduced by Pankov [20]

N ε ≔ { z ∈ E \ E − : ⟨ I ε ′ ( z ) , z ⟩ = 0 and ⟨ I ε ′ ( z ) , w ⟩ = 0 , ∀ w ∈ E − } ,

deeply studied by Szulkin and Weth [25]. Following the terminology of Szulkin and Weth [25], the set N ε is called the generalized Nehari manifold, it contains all nontrivial critical points of I ε . Let us denote by c ε the energy value defined by

c ε ≔ inf N ε I ε .

If c ε is achieved by z ε ∈ N ε , then z ε is called a ground state solution of system (2.1).

Furthermore, for every z ∈ E \ E − , we also need to define the subspace

E ( z ) = E − ⊕ R z = E − ⊕ R z + ,

and the convex subset

E ^ ( z ) = E − ⊕ R + z = E − ⊕ R + z + .

We introduce a crucial estimate, which plays a key role in the method of the generalized Nehari manifold.

Lemma 2.3

Assume that z ∈ E , w ∈ E − , and t ⩾ 0 with z ≠ t z + w , then we have the following estimate:

I ε ( t z + w ) − I ε ( z ) < I ε ′ ( z ) , t 2 − 1 2 z + t w .

In particular, let z ∈ N ε , w ∈ E − , and t ⩾ 0 with z ≠ t z + w , there holds

I ε ( t z + w ) < I ε ( z ) .

Proof

Let z ∈ E , w ∈ E − , and t ⩾ 0 with z ≠ t z + w , computing directly, we have

I ε ( t z + w ) − I ε ( z ) − I ε ′ ( z ) , t 2 − 1 2 z + t w = − 1 2 ‖ w ‖ 2 + 1 2 ∫ R N V ( ε x ) ∣ w ∣ 2 d x + ∫ R N F ^ ( t , z , w ) d x ,

where

F ^ ( t , z , w ) ≔ h ( ∣ z ∣ ) z t 2 − 1 2 z + t w + H ( ∣ z ∣ ) − H ( ∣ t z + w ∣ ) .

On the one hand, using ( V 1 ) and (2.3) we deduce that

− ‖ w ‖ 2 + ∫ R N V ( ε x ) ∣ w ∣ 2 d x ⩽ − ‖ w ‖ 2 + V max ‖ w ‖ 2 2 < − ‖ w ‖ 2 + ‖ w ‖ 2 2 ⩽ 0 .

On the other hand, using conditions ( h 1 ) – ( h 3 ) and following the arguments explored in [36] (see also [32, Lemma 2.5]), we can verify that F ^ ( t , z , w ) < 0 for all t ⩾ 0 . Therefore, we obtain the first conclusion from the above estimate. Furthermore, we take z ∈ N ε and w ∈ E − , then we know that ⟨ I ε ′ ( z ) , z ⟩ = ⟨ I ε ′ ( z ) , w ⟩ = 0 . Evidently, we obtain the second conclusion.□

According to Lemma 2.3, we can obtain a direct result, that is, if z ∈ N ε , then z is the unique global maximum of I ε ∣ E ^ ( z ) .

Lemma 2.4

Assume that conditions ( V 1 ) and ( h 1 ) – ( h 3 ) hold. Then we have

  1. there exist two positive constants ϱ and α such that c ε = inf N ε I ε ⩾ inf S ϱ I ε ⩾ α > 0 , where S ϱ ≔ { z ∈ E + , ‖ z ‖ = ϱ } ;

  2. ‖ z + ‖ ⩾ max 1 − V max 1 + V max ‖ z − ‖ , 2 α 1 + V max > 0 for all z ∈ N ε .

Proof

(a) Let z ∈ E + , then from (2.3), (2.4), and (2.5) we infer that

I ε ( z ) = 1 2 ‖ z ‖ 2 + 1 2 ∫ R N V ( ε x ) ∣ z ∣ 2 d x − ∫ R N H ( ∣ z ∣ ) d x ⩾ 1 2 ‖ z ‖ 2 − 1 2 V max ‖ z ‖ 2 2 − ε ‖ z ‖ 2 2 − C ε ‖ z ‖ p p ⩾ 1 − V max 2 − ε ‖ z ‖ 2 − π p p C ε ‖ z ‖ p .

Since V max < 1 and p > 2 , we can find that there exist two positive constants ρ and α both independent of ε such that inf S ϱ I ε ⩾ α .

On the other hand, let z ∈ N ε , then there exists s > 0 such that s z + ∈ E ( z ) ∩ S ϱ , so from Lemma 2.3 we can conclude that

I ε ( z ) = max E ^ ( z ) I ε ⩾ I ε ( s z + ) ⩾ inf S ϱ I ε .

Therefore, we show that conclusion (a) holds.

(b) Let z ∈ N ε , using ( V 1 ) , (2.6), and conclusion (a), we can obtain

α ⩽ c ε ⩽ 1 2 ( ‖ z + ‖ 2 − ‖ z − ‖ 2 ) + 1 2 ∫ R N V ( ε x ) ∣ z ∣ 2 d x − ∫ R N H ( ∣ z ∣ ) d x ⩽ 1 2 ( ‖ z + ‖ 2 − ‖ z − ‖ 2 ) + 1 2 ∫ R N V ( ε x ) ∣ z ∣ 2 d x ⩽ 1 2 ( ( 1 + V max ) ‖ z + ‖ 2 − ( 1 − V max ) ‖ z − ‖ 2 ) .

Thereby, it follows that

‖ z + ‖ ⩾ max 1 − V max 1 + V max ‖ z − ‖ , 2 α 1 + V max > 0 ,

completing the proof.□

Lemma 2.5

If W ⊂ E + \ { 0 } is a compact subset, then there is R > 0 such that I ε < 0 on E ^ ( z ) \ B R ( 0 ) for every z ∈ W .

Proof

Without loss of generality, we can assume that ‖ z ‖ = 1 for all z ∈ W . Suppose by contradiction that there exist { z n } ⊂ W and w n ∈ E ^ ( z n ) satisfying I ε ( w n ) ⩾ 0 for all n and ‖ w n ‖ → ∞ as n → ∞ . As W is compact, up to a subsequence, z n → z ∈ W with ‖ z ‖ = 1 .

Setting u n = w n ∕ ‖ w n ‖ = s n z n + u n − , where s n ⩾ 0 and u n − ∈ E − . Obviously, we have ‖ u n ‖ 2 = s n 2 ‖ z n ‖ 2 + ‖ u n − ‖ 2 = 1 . Passing to a subsequence, u n ⇀ u in E , u n − ⇀ u − in E , s n → s ⩾ 0 , and u n ( x ) → u ( x ) a.e. on R N . From ( V 1 ) , (2.3), and (2.6) we can derive that

0 ⩽ I ε ( w n ) ‖ w n ‖ 2 = 1 2 ( s n 2 ‖ z n ‖ 2 − ‖ u n − ‖ 2 ) + 1 2 ∫ R N V ( ε x ) ∣ u n ∣ 2 d x − ∫ R N H ( ∣ w n ∣ ) ∣ w n ∣ 2 ∣ u n ∣ 2 d x ⩽ 1 + V max 2 s n 2 ‖ z n ‖ 2 − 1 − V max 2 ‖ u n − ‖ 2 − ∫ R N H ( ∣ w n ∣ ) ∣ w n ∣ 2 ∣ u n ∣ 2 d x ⩽ 1 + V max 2 s n 2 ‖ z n ‖ 2 − 1 − V max 2 ‖ u n − ‖ 2 ,

which yields that s > 0 , and so u = s z + u − ≠ 0 . If this is not true, then s = 0 . From the above inequality we can see that ‖ u n − ‖ 2 → 0 , which contradicts with ‖ u n ‖ = 1 .

Letting Ω 0 ≔ { x ∈ R N : u ( x ) ≠ 0 } . Then ∣ Ω 0 ∣ > 0 , and ∣ w n ( x ) ∣ → ∞ in Ω 0 . Thereby, using ( f 3 ) and Fatou’s lemma we obtain

0 ⩽ lim n → ∞ I ε ( w n ) ‖ w n ‖ 2 = lim n → ∞ 1 2 ( s n 2 ‖ z n ‖ 2 − ‖ u n − ‖ 2 ) + 1 2 ∫ R N V ( ε x ) ∣ u n ∣ 2 d x − ∫ R N H ( ∣ w n ∣ ) ∣ w n ∣ 2 ∣ u n ∣ 2 d x ⩽ 1 + V max 2 s 2 − ∫ Ω 0 liminf n → ∞ H ( ∣ w n ∣ ) ∣ w n ∣ 2 ∣ u n ∣ 2 d x = − ∞ ,

which is absurd. The proof is completed.□

For the purpose of later proof, we need to prove the equivalent relationship between two norms. According to ( V 1 ) and (2.3), it is easy to obtain the following estimates:

(2.7) ( 1 − V max ) ‖ z + ‖ 2 ⩽ ‖ z + ‖ 2 ± V max ‖ z + ‖ 2 2 ⩽ ( 1 + V max ) ‖ z + ‖ 2

and

(2.8) ( 1 − V max ) ‖ z − ‖ 2 ⩽ ‖ z − ‖ 2 ± V max ‖ z − ‖ 2 2 ⩽ ( 1 + V max ) ‖ z − ‖ 2 .

Evidently, (2.7) and (2.8) yield that the norm

(2.9) ‖ z ‖ * = ( ‖ z ‖ 2 − V max ‖ z ‖ 2 2 ) 1 ∕ 2 ∼ ‖ z ‖ on E ,

where the symbol ∼ denotes the equivalence of two norms.

Lemma 2.6

For each z ∉ E − , the set N ε ∩ E ^ ( z ) consists of precisely one point m ^ ε ( z ) , which is the unique global maximum of I ε ∣ E ^ ( z ) . In other words, there exist unique t ε > 0 and w ε ∈ E − such that m ^ ε ( z ) = t ε z + + w ε ∈ N ε and

I ε ( m ^ ε ( z ) ) = max t ⩾ 0 , w ∈ E − I ε ( t z + + w ) .

Moreover, if z ∈ N ε , then t ε = 1 and w ε = z − .

Proof

We follow some ideas found in [25]. Indeed, according to Lemma 2.3, it suffices to show that N ε ∩ E ^ ( z ) ≠ ∅ . Since E ^ ( z ) = E − ⊕ R + z = E − ⊕ R + z + = E ^ ( z + ) , we may assume that z ∈ E + . From Lemma 2.5, we find that there exists R > 0 such that I ε ( z ) ⩽ 0 for z ∈ E ^ ( z ) \ B R ( 0 ) . Moreover, Lemma 2.4 implies that I ε ( t z ) > 0 for small t > 0 . Hence, 0 < sup I ε ( E ^ ( z ) ) < ∞ . If I ε is weakly upper semi-continuous on E ^ ( z ) , then we can find a z ^ ∈ E ^ ( z ) such that I ε ( z ^ ) = sup I ε ( E ^ ( z ) ) , and z ^ is a critical point of I ε ∣ E ^ ( z ) . Thereby, ⟨ I ε ′ ( z ^ ) , z ^ ⟩ = ⟨ I ε ′ ( z ^ ) , w ⟩ = 0 for all w ∈ E ^ ( z ) , which shows that z ^ ∈ N ε ∩ E ^ ( z ) .

Next we need to show that I ε is weakly upper semicontinuous on E ^ ( z ) . Let z n ⇀ z 0 in E with z n = t n z + z n − and z 0 = t 0 z + z 0 − , then t n → t 0 and z n − ⇀ z 0 − in E − . Hence, using (2.8), Fatou’s lemma and the weak lower semicontinuity of norm we can infer that

− I ε ( z 0 ) = 1 2 ( ‖ z 0 − ‖ 2 − t 0 2 ‖ z ‖ 2 ) − 1 2 ∫ R N V max ∣ t 0 z + z 0 − ∣ 2 d x + Γ ε ( z 0 ) = 1 2 ( ( ‖ z 0 − ‖ 2 − V max ‖ z 0 − ‖ 2 2 ) − t 0 2 ( ‖ z ‖ 2 + V max ‖ z ‖ 2 2 ) ) + Γ ε ( z 0 ) ⩽ liminf n → ∞ 1 2 ( ( ‖ z n − ‖ 2 − V max ‖ z n − ‖ 2 2 ) − t n 2 ( ‖ z ‖ 2 + V max ‖ z ‖ 2 2 ) ) + Γ ε ( z n ) = liminf n → ∞ [ − I ε ( z n ) ] ,

where we use the following fact:

Γ ε ( z ) = ∫ R N 1 2 ( V max − V ( ε x ) ) ∣ z ∣ 2 + H ( ∣ z ∣ ) d x ⩾ 0 .

This shows that I ε is weakly upper semicontinuous on E ^ ( z ) . The proof is completed.□

We point out that, as a consequence of Lemma 2.6, the ground state energy value c ε has a minimax characterization given by

(2.10) c ε = inf w ∈ E \ E − max z ∈ E ^ ( w ) I ε ( z ) = inf w ∈ E + \ { 0 } max z ∈ E ^ ( w ) I ε ( z ) .

Lemma 2.7

I ε is coercive on N ε , i.e., I ε ( z ) → ∞ as ‖ z ‖ → ∞ , z ∈ N ε .

Proof

Suppose by contradiction that there is a sequence { z n } ⊂ N ε such that ‖ z n ‖ → ∞ and I ε ( z n ) ⩽ c 0 for some c 0 ∈ [ α , + ∞ ) . Letting u n = z n ∕ ‖ z n ‖ , passing to a subsequence, then u n ⇀ u in E and u n ( x ) → u ( x ) a.e. on R N . It follows from Lemma 2.4 that

(2.11) ‖ u n + ‖ 2 = ‖ z n + ‖ 2 ‖ z n + ‖ 2 + ‖ z n − ‖ 2 ⩾ ‖ z n + ‖ 2 ‖ z n + ‖ 2 + 1 + V max 1 − V max ‖ z n + ‖ 2 = 1 − V max 2 .

On account of Lions’ concentration compactness principle, in the following we will discuss two cases: vanishing or nonvanishing.

If { u n + } is vanishing, the vanishing lemma found in [26, Lemma 2.1] yields that u n + → 0 in L q for all q ∈ ( 2 , 2 * ) . Therefore, for any s > 0 , we deduce from (2.5) that

(2.12) ∫ R N H ( ∣ s u n + ∣ ) d x → 0 .

Since s u n + ∈ E ^ ( z n ) for s ⩾ 0 , then using Lemma 2.3, (2.11), and (2.12) we can obtain

c 0 ⩾ I ε ( z n ) ⩾ I ε ( s u n + ) = s 2 2 ‖ u n + ‖ 2 + s 2 2 ∫ R N V ( ε x ) ∣ u n + ∣ 2 d x − ∫ R N H ( ∣ s u n + ∣ ) d x ⩾ 1 − V max 2 s 2 ‖ u n + ‖ 2 − ∫ R N H ( ∣ s u n + ∣ ) d x ⩾ 1 − V max 2 2 s 2 − ∫ R N H ( ∣ s u n + ∣ ) d x → 1 − V max 2 2 s 2 ,

which is absurd if s is large enough. So the vanishing case does not occur.

If { u n + } is nonvanishing, then there exist r , δ > 0 and a sequence { y n } ⊂ R N such that

∫ B r ( y n ) ∣ u n + ∣ 2 d x ⩾ δ .

Let us consider the sequence u ˜ n ( x ) = u n ( x + y n ) , then it follows that

∫ B r ( 0 ) ∣ u ˜ n + ∣ 2 d x ⩾ δ .

Clearly, we can find a u ˜ ∈ E such that u ˜ n + → u ˜ + in L loc 2 with u ˜ + ≠ 0 .

Setting Ω 1 ≔ { x ∈ R N : u ˜ ( x ) ≠ 0 } . Then ∣ Ω 1 ∣ > 0 and for each x ∈ Ω 1 , ∣ z n ( x + y n ) ∣ = ∣ u ˜ n ( x ) ∣ ‖ z n ‖ → ∞ . Therefore, employing ( h 2 ) and Fatou’s lemma we can derive that

0 ⩽ I ε ( z n ) ‖ z n ‖ 2 = 1 2 ( ‖ u n + ‖ 2 − ‖ u n − ‖ 2 ) + 1 2 ∫ R N V ( ε x ) ∣ u n ∣ 2 d x − ∫ R N H ( ∣ z n ∣ ) ‖ z n ‖ 2 d x ⩽ 1 + V max 2 ‖ u n + ‖ 2 − 1 − V max 2 ‖ u n − ‖ 2 − ∫ R N H ( ∣ z n ∣ ) ‖ z n ‖ 2 d x ⩽ 1 + V max 2 ‖ u n + ‖ 2 − 1 − V max 2 ‖ u n − ‖ 2 − ∫ Ω 1 H ( ∣ z n ( x + y n ) ∣ ) ∣ z n ( x + y n ) ∣ 2 ∣ u ˜ n ∣ 2 d x → − ∞ .

Evidently, we obtain a concentration. So, we finish the proof of lemma.□

Now we are going to verify the continuity of the map m ^ ε given in Lemma 2.6.

Lemma 2.8

The map m ^ ε : E + \ { 0 } → N ε is continuous.

Proof

We will adopt the similar arguments as in the proof of [25, Lemma 2.8] to prove the conclusion. Let z ∈ E + \ { 0 } , in view of a standard argument, the continuity of m ^ ε in z is reduced to the following assertion:

if z n → z for a sequence { z n } ⊂ E + \ { 0 } , then m ^ ε ( z n ) → m ^ ε ( z ) .

Without loss of generality, we may assume that ‖ z n ‖ = ‖ z ‖ = 1 for all n , so m ^ ε ( z n ) = ‖ m ^ ε ( z n ) + ‖ z n + m ^ ε ( z n ) − . By Lemma 2.5, we find that there exists R > 0 such that

I ε ( m ^ ε ( z n ) ) = sup I ε ( E ( z n ) ) ⩽ sup I ε ( B R ( 0 ) ) ⩽ sup u ∈ B R ( 0 ) ‖ z + ‖ = R 2 , ∀ n ∈ N .

Therefore, m ^ ε ( z n ) is bounded by Lemma 2.7. Up to a subsequence, we can assume that

t n ≔ ‖ m ^ ε ( z n ) + ‖ → t and m ^ ε ( z n ) − ⇀ z ∗ − in E .

where t > 0 by Lemma 2.4. Furthermore, using Lemma 2.6 we can infer that

limsup n → ∞ I ε ( m ^ ε ( z n ) ) ⩾ limsup n → ∞ I ε ( s z n + w ) ⩾ I ε ( s z + w ) , ∀ s ⩾ 0 , w ∈ E − .

Applying Fatou’s lemma and the weak lower semicontinuity of norm, and combining (2.9) we can conclude that

I ε ( t z + z ∗ − ) ⩽ limsup n → ∞ I ε ( m ^ ε ( z n ) ) = limsup n → ∞ t n 2 2 − 1 2 ‖ m ^ ε ( z n ) − ‖ 2 + 1 2 ∫ R N V ( ε x ) ∣ m ^ ε ( z n ) ∣ 2 d x − ∫ R N H ( ∣ m ^ ε ( z n ) ∣ ) d x = limsup n → ∞ t n 2 2 − 1 2 ‖ m ^ ε ( z n ) − ‖ 2 − ∫ R N H ( ∣ m ^ ε ( z n ) ∣ ) d x + t n 2 2 ∫ R N V ( ε x ) ∣ z n ∣ 2 d x + 1 2 ∫ R N V ( ε x ) ∣ m ^ ε ( z n ) − ∣ 2 d x ⩽ t 2 2 − 1 2 ‖ z ∗ − ‖ 2 + t 2 2 ∫ R N V ( ε x ) ∣ z ∣ 2 d x + 1 2 ∫ R N V ( ε x ) ∣ z ∗ − ∣ 2 d x − ∫ R N H ( ∣ t z + z ∗ − ∣ ) d x = I ε ( t z + z ∗ − ) ,

which implies that m ^ ε ( z n ) − → z ∗ − in E . Moreover, the argument above also yields that

I ε ( t z + z ∗ − ) ⩾ I ε ( s z + w ) , ∀ s ⩾ 0 , w ∈ E − ,

which shows that m ^ ε ( z ) = t z + z ∗ − . Therefore, we can see that

m ^ ε ( z n ) → m ^ ε ( z ) in E .

So the assertion follows, completing the proof.□

Based on the comments above, next we will introduce some important results of the method of the generalized Nehari manifold. To do this, we set S + ≔ { z ∈ E + : ‖ z ‖ = 1 } in E + and consider the following maps

m ^ ε : E + \ { 0 } → N ε and m ε = m ^ ε ∣ S + ,

and the inverse of m ε is

m ε − 1 : N ε → S + , m ε − 1 ( z ) = z + ∕ ‖ z + ‖ .

From Lemma 2.8, it is not difficult to see that m ε is a homeomorphism.

From now on, let us consider the reduction functional

Φ ^ ε ( z ) = I ε ( m ^ ε ( z ) ) and Φ ε = Φ ^ ε ∣ S + .

Evidently, Lemma 2.8 shows that they are continuous.

The next results establish some crucial properties involving the reduced functionals Φ ^ ε and Φ ε , which play a fundamental role in the study of the existence of ground state solutions for strongly indefinite variational problems. And their proofs follow the proofs of [25, Proposition 2.9, Corollary 2.10].

Lemma 2.9

We have the following properties:

  1. Φ ^ ε ∈ C 1 ( E + \ { 0 } , R ) and for z , w ∈ E + and z ≠ 0 ,

    ⟨ Φ ^ ε ′ ( z ) , w ⟩ = ‖ m ^ ε ( z ) + ‖ ‖ z ‖ ⟨ I ε ′ ( m ^ ε ( z ) ) , w ⟩ .

  2. Φ ε ∈ C 1 ( S + , R ) and for each z ∈ S + and w ∈ T z ( S + ) = { v ∈ E + : ( z , v ) = 0 } ,

    ⟨ Φ ε ′ ( z ) , w ⟩ = ‖ m ^ ε ( z ) + ‖ ⟨ I ε ′ ( m ^ ε ( z ) ) , w ⟩ .

  3. { z n } is a (PS)-sequence for Φ ε if and only if { m ^ ε ( z n ) } is a (PS)-sequence for I ε .

  4. z ∈ S + is a critical point of Φ ε if and only if m ^ ε ( z ) ∈ N ε is a critical point of I ε . Moreover, the corresponding values of Φ ε and I ε coincide and

    c ε = inf S + Φ ε = inf N ε I ε .

3 Autonomous problem

We will draw upon the techniques of the limit problem to help us to prove the main result. To this end, in this section we need to study the existence and some properties of the ground state solutions for the constant coefficient system.

Let κ ∈ ( − 1 , 1 ) , we start by considering the autonomous problem:

(3.1) − Δ u + b → ⋅ ∇ u + u + κ v = H v ( u , v ) in R N , − Δ v − b → ⋅ ∇ v + v + κ u = H u ( u , v ) in R N .

It is well known that the solutions of problem (3.1) are precisely critical points of the energy functional defined by

I κ ( z ) = 1 2 ( ‖ z + ‖ 2 − ‖ z − ‖ 2 ) + κ 2 ∫ R N ∣ z ∣ 2 d x − ∫ R N H ( ∣ z ∣ ) d x .

Similar to the previous section, we define the associated generalized Nehari manifold

N κ ≔ { z ∈ E \ E − : ⟨ I κ ′ ( z ) , z ⟩ = 0 and ⟨ I κ ′ ( z ) , w ⟩ = 0 , ∀ w ∈ E − }

and the ground state energy value

c κ = inf N κ I κ .

Employing the same arguments used in the previous section, we can know that for every z ∈ E \ E − , the set N κ ∩ E ^ ( z ) is a singleton set, and the element of this set is the unique global maximum of I κ ∣ E ^ ( z ) , that is, there exists a unique pair t > 0 and w ∈ E − such that

I κ ( t z + w ) = max u ∈ E ^ ( z ) I κ ( u ) .

Accordingly, we can define the maps

m ^ κ : E + \ { 0 } → N κ and m κ = m ^ κ ∣ S + ,

and the inverse of m κ is

m κ − 1 : N κ → S + , m κ − 1 ( z ) = z + ∕ ‖ z + ‖ .

Meanwhile, we consider the reduction functional Φ ^ κ : E + \ { 0 } → R and the restriction Φ κ : S + → R

Φ ^ κ ( z ) = I κ ( m ^ κ ( z ) ) and Φ κ = Φ ^ κ ∣ S + .

We would like to clarify that, using same discussions explored in Section 2, all related conclusions and properties in Section 2 remain for I κ , c κ , N κ , m ^ κ , m κ , Φ ^ κ , and Φ κ , respectively. Here we omit the details of proof.

Moreover, we also have a minimax characterization for ground state energy value c κ

c κ = inf w ∈ E \ E − max z ∈ E ^ ( w ) I κ ( z ) = inf w ∈ E + \ { 0 } max z ∈ E ^ ( w ) I κ ( z ) = inf S + Φ κ .

We now state the existence result of ground state solution for the autonomous problem (3.1).

Lemma 3.1

Assume that κ ∈ ( − 1 , 1 ) and ( h 1 ) – ( h 3 ) hold, then problem (3.1) possesses a ground state solution z such that I κ ( z ) = c κ > 0 .

Proof

First, it follows from Lemma 2.4 that c κ > 0 . We observe that if z ∈ N κ satisfies I κ ( z ) = c κ , then from Lemma 2.9 we can see that

Φ κ ( m κ − 1 ( z ) ) = I κ ( z ) = c κ = inf S + Φ κ .

This shows that m κ − 1 ( z ) ∈ S + is a minimizer of Φ κ , and hence a critical point of Φ κ , so that z is a critical point of I κ by Lemma 2.9.

Thereby, it remains to prove that there exists a minimizer z ˜ ∈ N κ such that I κ ( z ˜ ) = c κ . In fact, using Ekeland’s variational principle [26], there exists a sequence { w n } ⊂ S + such that Φ κ ( w n ) → c κ and Φ κ ′ ( w n ) → 0 as n → ∞ . Let z n = m ^ κ ( w n ) ∈ N κ for all n ∈ N . Then Lemma 2.9 shows that I κ ( z n ) → c κ and I κ ′ ( z n ) → 0 . On account of Lemma 2.7, it is easy to see that { z n } is bounded in E and hence z n ⇀ z in E for some z ∈ E after passing to a subsequence.

Let { y n } ⊂ R N be a sequence satisfying

∫ B 1 ( y n ) ∣ z n ∣ 2 d x = max y ∈ R N ∫ B 1 ( y ) ∣ z n ∣ 2 d x .

Using the fact that I κ and N κ are invariant under translations, we may assume that { y n } is bounded in R N . If

(3.2) lim n → ∞ ∫ B 1 ( y n ) ∣ z n ∣ 2 d x = 0 ,

then, Lions’ concentration compactness principle [26, Lemma 2.1] yields that z n → 0 in L q for any q ∈ ( 2 , 2 * ) . From (2.5) we can conclude that

∫ R N 1 2 h ( ∣ z n ∣ ) ∣ z n ∣ 2 − H ( ∣ z n ∣ ) d x = o n ( 1 ) ,

and it follows that

c κ + o n ( 1 ) = I κ ( z n ) − 1 2 ⟨ I ′ ( z n ) , z n ⟩ = ∫ R N 1 2 h ( ∣ z n ∣ ) ∣ z n ∣ 2 − H ( ∣ z n ∣ ) d x = o n ( 1 ) .

This is impossible since c κ > α > 0 , and it follows that (3.2) cannot hold.

Let us define z ˜ n ( x ) = z n ( x + y n ) , then we directly obtain ‖ z ˜ n ‖ = ‖ z n ‖ and

(3.3) ∫ B 1 ( 0 ) ∣ z ˜ n ∣ 2 d x > 0 .

Moreover, there holds

(3.4) I κ ( z ˜ n ) → c κ and I ′ ( z ˜ n ) → 0 .

Therefore, we can find a z ˜ ∈ E such that z ˜ n ⇀ z ˜ in E , z ˜ n → z ˜ in L loc p for all p ∈ [ 2 , 2 * ) , and z ˜ n ( x ) → z ˜ ( x ) a.e. on R N . From (3.3) and (3.4) we can infer that z ˜ ≠ 0 , I ′ ( z ˜ ) = 0 .

Finally, we will show that I κ ( z ˜ ) = c κ . Exploiting Fatou’s lemma and (2.6), we can derive that

c κ = lim n → ∞ I κ ( z ˜ n ) − 1 2 ⟨ I ′ ( z ˜ n ) , z ˜ n ⟩ = lim n → ∞ ∫ R N 1 2 h ( ∣ z ˜ n ∣ ) ∣ z ˜ n ∣ 2 − H ( ∣ z ˜ n ∣ ) d x ⩾ ∫ R N lim n → ∞ 1 2 h ( ∣ z ˜ n ∣ ) ∣ z ˜ n ∣ 2 − H ( ∣ z ˜ n ∣ ) d x = I κ ( z ˜ ) − 1 2 ⟨ I ′ ( z ˜ ) , z ˜ ⟩ = I κ ( z ˜ ) ,

which implies that I κ ( z ˜ ) ⩽ c κ . The reverse inequality follows from the definition of c κ since z ˜ ∈ N κ . So, z ˜ is a ground state solution of problem (3.1), completing the proof.□

As a byproduct of Lemma 3.1, we obtain the conclusion involving the monotonicity and continuity of c κ .

Lemma 3.2

The function κ ↦ c κ is strictly increasing and continuous on ( − 1 , 1 ) .

Proof

In what follows, let z κ 1 and z κ 2 be as ground state solution of I κ 1 and I κ 2 . Assume that − 1 < κ 1 < κ 2 < 1 . First of all, we prove that the function κ ↦ c κ is increasing. By Lemma 2.6, there exist t 1 > 0 and w 1 ∈ E − such that

I κ 1 ( t 1 z κ 2 + w 1 ) = max z ∈ E ^ ( z κ 2 ) I κ 1 ( z ) ,

then it follows that

c κ 1 ⩽ I κ 1 ( t 1 z κ 2 + w 1 ) = I κ 2 ( t 1 z κ 2 + w 1 ) + κ 1 − κ 2 2 ∫ R N ∣ t 1 z κ 2 + w 1 ∣ 2 d x ⩽ I κ 2 ( z κ 2 ) + κ 1 − κ 2 2 ∫ R N ∣ t 1 z κ 2 + w 1 ∣ 2 d x = c κ 2 + κ 1 − κ 2 2 ∫ R N ∣ t 1 z κ 2 + w 1 ∣ 2 d x .

Since κ 2 − κ 1 > 0 , it is easy to see that

c κ 1 < c κ 2 .

This shows that the function κ ↦ c κ is strictly increasing on ( − 1 , 1 ) . Next we will take two cases to complete the proof of the continuity of c κ .

Case 1: Let { κ n } be a sequence such that κ 1 ⩾ κ 2 ⩾ ⋯ ⩾ κ n ⩾ ⋯ ⩾ κ and κ n → κ .

Claim: c κ n → c κ as n → ∞ .

Indeed, let z κ be the ground state solution of problem (3.1). In view of Lemma 2.6, we can find that there exist t n > 0 and w n ∈ E − such that

I κ n ( t n z κ + w n ) = max z ∈ E ^ ( z κ ) I κ n ( z ) for all n ∈ N .

Computing directly, we have

I κ 1 ( z ) − I κ n ( z ) = ( κ 1 − κ n ) ∫ R N ∣ z ∣ 2 d x ⩾ 0 , for n ∈ N and z ∈ E ,

which yields that I κ 1 ( z ) ⩾ I κ n ( z ) . From Lemma 2.5, there exists R > 0 such that

(3.5) I κ n ( z ) ⩽ I κ 1 ( z ) ⩽ 0 , ∀ z ∈ E ^ ( z κ ) \ B R ( 0 ) .

According to the monotonicity of c κ , we immediately obtain

I κ n ( t n z κ + w n ) = max z ∈ E ^ ( z κ ) I κ n ( z ) ⩾ c κ n ⩾ c κ > 0 ,

which together with (3.5), yields that ‖ t n z κ + w n ‖ ⩽ R . Therefore, { s n z μ + φ n } is bounded in E . Then, we can derive that

c κ n ⩽ I κ n ( t n z κ + w n ) = I κ ( t n z κ + w n ) + κ n − κ 2 ∫ R N ∣ t n z κ + w n ∣ 2 d x ⩽ I κ ( z κ ) + κ n − κ 2 ∫ R N ∣ t n z κ + w n ∣ 2 d x = c κ + o n ( 1 ) .

On the other hand, since c κ ⩽ c κ n for all n ∈ N , then we show that

c κ n → c κ as n → ∞ .

Case 2: Let { κ n } be a sequence such that κ 1 ⩽ κ 2 ⩽ ⋯ ⩽ κ n ⩽ ⋯ ⩽ κ and κ n → κ .

Claim: c κ n → c κ as n → ∞ .

Indeed, let z n be the ground state solution of problem (3.1) with κ = κ n , then there exist s n > 0 and w n ∈ E − such that

I κ ( s n z n + w n ) = max z ∈ E ^ ( z n ) I κ ( z ) .

Similar to the previous argument, we can easily obtain that the sequence { z n } is bounded. Moreover, we can find that there exist δ > 0 , r > 0 and { y n } ⊂ R N such that for each n ∈ N , we have

(3.6) ∫ B r ( y n ) ∣ z n + ∣ 2 d x ⩾ δ .

If not, using Lions’ concentration compactness principle we have z n + → 0 in L q ( R ) for all q ∈ [ 2 , 2 * ) . Then we can check that

∫ R N h ( ∣ z n ∣ ) z n z n + d x → 0 .

Thereby, it follows that

0 = ⟨ I κ n ′ ( z n ) , z n + ⟩ = ‖ z n + ‖ 2 + κ n ∫ R N z n z n + d x − ∫ R N h ( ∣ z n ∣ ) z n z n + d x ⩾ ( 1 − ∣ κ n ∣ ) ‖ z n + ‖ 2 + o n ( 1 ) ,

which shows that ‖ z n + ‖ 2 → 0 since κ n ∈ ( − 1 , 1 ) , contradicting the fact ‖ z n + ‖ 2 ⩾ 2 c κ n > 0 . So, (3.6) holds.

Setting z ˜ n ( x ) ≔ z n ( x + y n ) , one can check that { z ˜ n } is bounded, up to a subsequence, z ˜ n + ⇀ z ˜ + ≠ 0 in E . Denote V ≔ { z ˜ n + } ⊂ E + \ { 0 } , hence V is bounded and the sequence does not weakly converge to zero in E . Then using Lemma 2.5, there exists R > 0 such that for every z ∈ V , we obtain

(3.7) I κ ( w ) < 0 , for w ∈ E ( z ) \ B R ( 0 ) .

Let us define w ˜ n ( x ) ≔ w n ( x + y n ) , we have

(3.8) I κ ( s n z ˜ n + w ˜ n ) = I κ ( s n z n + w n ) = max z ∈ E ^ ( z n ) I κ ( z ) ⩾ c κ > 0 , ∀ n ∈ N .

Gathering (3.7) and (3.8), we know ‖ s n z ˜ n + w ˜ n ‖ ⩽ R for all n ∈ N , then ‖ s n z n + w n ‖ ⩽ R , which implies that the sequence { s n z n + w n } is bounded in E . Then it follows that

c κ ⩽ I κ ( s n z n + w n ) = I κ n ( s n z n + w n ) + κ − κ n 2 ∫ R N ∣ s n z n + w n ∣ 2 d x ⩽ I κ n ( z n ) + κ − κ n 2 ∫ R N ∣ s n z n + w n ∣ 2 d x = c κ n + o n ( 1 ) .

Combining this with the fact that c κ ⩾ c κ n for all n ∈ N , we show that

c κ n → c κ as n → ∞ .

This completes the proof.□

Finally, in order to conclude this section, we establish a compactness result which will be useful in our analysis.

Lemma 3.3

Let { z n } be a Palais-Smale sequence at level c κ for I κ with z n ⇀ z in E . Then, we have the following conclusions:

  1. z n → z in E ; or

  2. there exists { y n } ⊂ R N with ∣ y n ∣ → ∞ such that the sequence z ^ n ( x ) = z n ( x + y n ) is strongly convergent to z ^ in E for some z ^ ∈ E \ { 0 } .

Proof

We start the proof by showing that if z ≠ 0 , then conclusion (a) is valid. Indeed, if { z n } is a Palais-Smale sequence at level c κ for I κ , then it is easy to see that I κ ′ ( z ) = 0 . If z ≠ 0 , it follows that z ∈ N κ and

(3.9) c κ ⩽ I κ ( z ) = I κ ( z ) − 1 2 ⟨ I κ ′ ( z ) , z ⟩ = ∫ R N 1 2 h ( ∣ z ∣ ) ∣ z ∣ 2 − H ( ∣ z ∣ ) d x ⩽ liminf n → ∞ ∫ R N 1 2 h ( ∣ z n ∣ ) ∣ z n ∣ 2 − H ( ∣ z n ∣ ) d x ⩽ limsup n → ∞ I κ ( z n ) − 1 2 ⟨ I κ ′ ( z n ) , z n ⟩ = c κ .

Following some ideas of proof [11, Lemma 6.7], we can check that

I κ ( z n − z ) → c κ − I κ ( z ) and I κ ′ ( z n − z ) → 0 .

This, together with (3.9), yields that I κ ( z n − z ) → 0 and I κ ′ ( z n − z ) → 0 . Using the fact that the zero function 0 is an isolated critical point of I κ , we conclude that z n → z in E .

Now we are going to verify that if z = 0 , then conclusion (b) holds. In fact, if z = 0 , arguing as the previous result, we can find that there exist δ > 0 , r > 0 and { y n } ⊂ R N such that

∫ B r ( y n ) ∣ z n + ∣ 2 d x ⩾ δ .

Since z n + ⇀ 0 in E , it is not difficult to see that { y n } is an unbounded sequence. Setting z ^ n ( x ) = z n ( x + y n ) , then I κ ( z n ) = I κ ( z ^ n ) → c κ and I κ ′ ( z n ) = I κ ′ ( z ^ n ) → 0 . It follows that { z ^ n } is also a Palais-Smale sequence at level c κ for I κ with z ^ n ⇀ z ^ in E and z ^ + ≠ 0 , that is, z ^ ≠ 0 . Repeating the above arguments explored in (a), we can deduce that z ^ n → z ^ in E , completing the proof.□

4 Proof of Theorem 1.1

In the section, we establish some important results and give the proof of the multiplicity result of semiclassical solutions for system (1.1).

To prove the main results, we will use some conclusions of limit problem. To do this, we consider the following limit system:

(4.1) − Δ u + b → ⋅ ∇ u + u + V ( 0 ) v = H v ( u , v ) in R N , − Δ v − b → ⋅ ∇ v + v + V ( 0 ) u = H u ( u , v ) in R N .

For convenience, next we will use the notations I V ( 0 ) , c V ( 0 ) , and N V ( 0 ) to denote the associated energy functional, ground state energy value, and Nehari manifold of system (4.1), respectively.

Next, we introduce the relationship of the ground state energy value between system (2.1) and limit system (4.1), and this is very crucial in our following arguments.

Lemma 4.1

We have the limit lim ε → 0 c ε = c V ( 0 ) .

Proof

Let ε n → 0 as n → ∞ . Clearly, using Lemma 3.2 we know that c V ( 0 ) ⩽ c ε n for all n ∈ N , and so c V ( 0 ) ⩽ liminf n → ∞ c ε n .

On the other hand, from Lemma 3.1 we can see that system (4.1) has a ground state solution z 0 . Then using Lemma 2.6 we can find that there exist t n > 0 and w n ∈ E − such that t n z 0 + + w n ∈ N ε n and

I ε n ( t n z 0 + + w n ) ⩾ c ε n ⩾ c V ( 0 ) > 0 , ∀ n ∈ N .

Similar to the previous arguments, we can know that { t n z 0 + + w n } is bounded in E . Thus, we may assume that t n → t 0 and w n ⇀ w in E − . Thereby, using (2.7) and (2.9) and applying the weakly lower semicontinuity of the norm and Fatou’s lemma we obtain

c V ( 0 ) = liminf n → ∞ c ε n ⩽ limsup n → ∞ c ε n ⩽ limsup n → ∞ I ε n ( t n z 0 + + w n ) = limsup n → ∞ 1 2 t n 2 ‖ z 0 + ‖ 2 − 1 2 ‖ w n ‖ 2 + V max 2 ∫ R N ∣ t n z 0 + + w n ∣ 2 d x − 1 2 ∫ R N [ V max − V ( ε n x ) ] ∣ t n z 0 + + w n ∣ 2 d x − ∫ R N H ( ∣ t n z 0 + + w n ∣ ) d x = limsup n → ∞ t n 2 2 ‖ z 0 + ‖ 2 + ∫ R N V max ∣ z 0 + ∣ 2 d x − 1 2 ‖ w n ‖ 2 − ∫ R N V max ∣ w n ∣ 2 d x − 1 2 ∫ R N [ V max − V ( ε n x ) ] ∣ t n z 0 + + w n ∣ 2 d x − ∫ R N H ( ∣ t n z 0 + + w n ∣ ) d x ⩽ t 0 2 2 ‖ z 0 + ‖ 2 + ∫ R N V max ∣ z 0 + ∣ 2 d x − 1 2 ‖ w ‖ 2 − ∫ R N V max ∣ w ∣ 2 d x − 1 2 ∫ R N [ V max − V ( 0 ) ] ∣ t 0 z 0 + + w ∣ 2 d x − ∫ R N H ( ∣ t 0 z 0 + + w ∣ ) d x = t 0 2 2 ‖ z 0 + ‖ 2 − 1 2 ‖ w ‖ 2 + V max 2 ∫ R N ∣ t 0 z 0 + + w ∣ 2 d x − ∫ R N H ( ∣ t 0 z 0 + + w ∣ ) d x = I V ( 0 ) ( t 0 z 0 + + w ) ⩽ I V ( 0 ) ( z 0 ) = c V ( 0 ) .

Evidently, we can obtain

lim n → ∞ c ε n = c V ( 0 ) ,

and we complete the proof.□

As a byproduct of Lemma 4.1, we can directly obtain the following result.

Lemma 4.2

Assume that condition ( V 1 ) holds, then there is ε 0 > 0 such that c ε < c V ∞ for ε ∈ ( 0 , ε 0 ) .

Proof

According to condition ( V 1 ) we know that V ( 0 ) < V ∞ . Then using Lemma 3.2 we have c V ( 0 ) < c V ∞ . It follows from Lemma 4.1 that there is ε 0 > 0 small enough such that c ε < c V ∞ for all ε ∈ ( 0 , ε 0 ) .□

From ( V 1 ) and ( V 2 ) we observe that V ( 0 ) = V 0 , then c V ( 0 ) = c V 0 . Next we establish compactness criteria for the functional Φ ε , which is crucial in our approach.

Lemma 4.3

The functional Φ ε satisfies the ( P S ) c compactness condition for c ⩽ c V 0 + π , where π = 1 2 ( c V ∞ − c V 0 ) .

Proof

Let { w n } ⊂ S + be a Palais-Smale sequence at level c for Φ ε with c ⩽ c V 0 + π . From Lemma 2.9, z n = m ε ( w n ) is a Palais-Smale sequence at level c for functional I ε . Then, { z n } is bounded in E , and passing to a subsequence, we may assume that z n ⇀ z in E and z n ( x ) → z ( x ) a.e. in R N .

Setting u n = z n − z , then similar to the proof of [34, Lemma 5.2], we obtain the following results:

(4.2) I ε ( u n ) = I ε ( z n ) − I ε ( z ) + o n ( 1 ) and I ε ′ ( u n ) = I ε ′ ( z n ) − I ε ′ ( z ) + o n ( 1 ) .

Observe that since ⟨ I ε ′ ( z n ) , ψ ⟩ → 0 , we obtain I ε ′ ( z ) = 0 . Moreover, using (2.6) we can deduce that

I ε ( z ) = I ε ( z ) − 1 2 ⟨ I ε ′ ( z ) , z ⟩ = ∫ R N 1 2 h ( ∣ z ∣ ) ∣ z ∣ 2 − H ( ∣ z ∣ ) d x ⩾ 0 .

This, together with (4.2), yields that

(4.3) I ε ′ ( u n ) → 0 and I ε ( u n ) → c ^ = c − I ε ( z ) .

Evidently, { u n } is a Palais-Smale sequence at level c ^ for I ε with c ^ ⩽ c V 0 + π .

Claim: for each R > 0 fixed, there holds

(4.4) lim n → ∞ sup y ∈ R N ∫ B R ( y ) ∣ u n ∣ 2 d x = 0 .

If Claim is true, then making use of the Lions concentration compactness principle, we can know that u n → 0 in L q for any q ∈ ( 2 , 2 * ) . From the facts that the orthogonal projection of E on E + and E − is continuous in L q , we have u n + → 0 and u n − → 0 in L q for any q ∈ ( 2 , 2 * ) . Thus, it follows from (2.5) that

(4.5) ∫ R N h ( ∣ u n ∣ ) u n ( u n + − u n − ) d x → 0 .

Using (4.3) and (4.6) we can obtain

(4.6) ‖ u n ‖ 2 = ⟨ I ε ′ ( u n ) , u n + − u n − ⟩ + ∫ R N h ( ∣ u n ∣ ) u n ( u n + − u n − ) d x → 0 .

Therefore, we can see that u n → 0 , that is, z n → z in E .

In what follows, we will show that Claim is true. In fact, if Claim does not hold, there exist R > 0 , δ > 0 and { y n } ⊂ R N such that

(4.7) limsup n → ∞ ∫ B R ( y n ) ∣ u n ∣ 2 d x ⩾ δ .

This ensures that { y n } is an unbounded sequence since u n ⇀ 0 . Setting u ˜ n ( x ) = u n ( x + y n ) , then { u ˜ n } is bounded in E . By (4.7) we have

limsup n → ∞ ∫ B R ( 0 ) ∣ u ˜ n ∣ 2 d x ⩾ δ .

Thereby, we can find that there exists u ˜ ∈ E \ { 0 } such that u ˜ n ⇀ u ˜ in E and u ˜ n ( x ) → u ˜ ( x ) a.e. in R N .

On the other hand, according to the fact that ⟨ I ε ′ ( u n ) , ψ ( x + y n ) ⟩ = o n ( 1 ) for all ψ ∈ E , we can check that ⟨ I V ∞ ′ ( u ˜ ) , ψ ⟩ = 0 for all ψ ∈ E , which shows that u ˜ is a nontrivial critical point of I V ∞ . It then follows that

c V ∞ ⩽ I V ∞ ( u ˜ ) = I V ∞ ( u ˜ ) − 1 2 ⟨ I V ∞ ′ ( u ˜ ) , u ˜ ⟩ = ∫ R N 1 2 h ( ∣ u ˜ ∣ ) ∣ u ˜ ∣ 2 − H ( ∣ u ˜ ∣ ) d x ⩽ liminf n → ∞ ∫ R N 1 2 h ( ∣ u ˜ n ∣ ) ∣ u ˜ n ∣ 2 − H ( ∣ u ˜ n ∣ ) d x = liminf n → ∞ I ε ( u ˜ n ) − 1 2 ⟨ I ε ′ ( u ˜ n ) , u ˜ n ⟩ = c ^ ⩽ c V 0 + π .

This, together with the definition of π , yields that c V ∞ ⩽ c V 0 . Clearly, we obtain a contradiction since c V ∞ > c V 0 (see Lemma 3.2), completing the proof.□

Below let us fix ϱ 0 , r 0 > 0 such that

  • B ϱ 0 ( x i ) ¯ ∩ B ϱ 0 ( x j ) ¯ = ∅ for i ≠ j and i , j ∈ { 1 , … , k } ;

  • ⋃ i = 1 k B ϱ 0 ( x i ) ⊂ B r 0 ( 0 ) ;

  • L ϱ 0 = ⋃ i = 1 k B ϱ 0 ( x i ) ¯ .

Furthermore, we define the barycenter map Q ε : E \ { 0 } → R N by

Q ε ( u ) = ∫ R N ξ ( ε x ) ∣ u ∣ 2 d x ∫ R N ∣ u ∣ 2 d x ,

where ξ : R N → R N is given by

ξ ( x ) = x , if ∣ x ∣ ⩽ r 0 , x ∣ x ∣ r 0 , if ∣ x ∣ > r 0 .

Following the arguments in [2], we can prove the following important lemma, which is very useful to obtain ( P S ) c sequences for Φ ε .

Lemma 4.4

There exist π 0 > 0 and ε 0 > 0 such that if w ∈ S + and Φ ε ( w ) ⩽ c V 0 + π 0 , then Q ε ( w ) ∈ L ϱ 0 ∕ 2 for all ε ∈ ( 0 , ε 0 ) .

Proof

Assume by contradiction that there exist π n → 0 , ε n → 0 , and w n ∈ S + such that

Φ ε n ( w n ) ⩽ c V 0 + π n and Q ε n ( w n ) ∉ L ϱ 0 ∕ 2 .

By Lemma 2.6, there exist t n > 0 and φ n ∈ E − such that m ^ V 0 ( w n ) = t n w n + + φ n ∈ N V 0 . Then, for ∀ n ∈ N we have

c V 0 + π n ⩾ Φ ε n ( w n ) = I ε n ( m ε n ( w n ) ) ⩾ I ε n ( t n w n + + φ n ) ⩾ I V 0 ( t n w n + + φ n ) = I V 0 ( m ^ V 0 ( w n ) ) = Φ V 0 ( w n ) ⩾ c V 0 ,

which follows that

{ w n } ⊂ S + and Φ V 0 ( w n ) → c V 0 .

From Ekeland’s variational principle, we can assume that Φ V 0 ′ ( w n ) → 0 . Hence, we can see that z n = m ^ V 0 ( w n ) satisfies

{ z n } ⊂ N V 0 , I V 0 ( z n ) → c V 0 , and I V 0 ′ ( z n ) → 0 .

Making use of Lemma 3.3, there are two cases that need to be analyzed.

  1. z n → z ≠ 0 in E ; or

  2. there exists { y n } ⊂ R N with ∣ y n ∣ → ∞ such that the sequence z ^ n ( x ) = z n ( x + y n ) is strongly convergent to z ^ in E for some z ^ ∈ E \ { 0 } .

Evidently, if (a) holds, then we have w n → w ≠ 0 in E since m ^ V 0 is a homeomorphism from S + to N V 0 . But, if (b) holds, then we have w ^ n = w n ( x + y n ) → w ^ in E . Next we will use the following relation:

z ^ n ( x ) = z n ( x + y n ) = m ^ V 0 ( w n ( x + y n ) ) = m ^ V 0 ( w ^ n ( x ) ) .

Case (a): Thanks to the conclusion w n → w ≠ 0 in E , then applying the Lebesgue’s dominated convergence theorem we can deduce that

Q ε n ( w n ) = ∫ R N ξ ( ε n x ) ∣ w n ∣ 2 d x ∫ R N ∣ w n ∣ 2 d x → ∫ R N ξ ( 0 ) ∣ w ∣ 2 d x ∫ R N ∣ w ∣ 2 d x = 0 ∈ L ϱ 0 ∕ 2 .

This shows that Q ε n ( w n ) ∈ L ϱ 0 ∕ 2 for n large enough, and we obtain a contradiction.

Case (b): Setting z n = m ε n ( w n ) , z ^ n ( x ) = z n ( x + y n ) , and w ^ n = w n ( x + y n ) . Here there are two possibilities that need to be discussed: ∣ ε n y n ∣ → ∞ and ε n y n → y for some y ∈ R N .

If the case that ∣ ε n y n ∣ → ∞ does occur, in view of Lemma 2.6 we find that s > 0 and φ ^ ∈ E − such that s w ^ + + φ ^ ∈ N V ∞ ; it follows from w ^ n + → w ^ + in E that

c V ∞ ⩽ I V ∞ ( s w ^ + + φ ^ ) = s 2 2 ‖ w ^ + ‖ 2 − 1 2 ‖ φ ^ ‖ 2 + 1 2 ∫ R N V ∞ ∣ s w ^ + + φ ^ ∣ 2 d x − ∫ R N H ( ∣ s w ^ + + φ ^ ∣ ) d x = lim n → ∞ s 2 2 ‖ w ^ n + ‖ 2 − 1 2 ‖ φ ^ ‖ 2 − ∫ R N H ( ∣ s w ^ n + + φ ^ ∣ ) d x + 1 2 ∫ R N V ( ε n ( x + y n ) ) ∣ s w ^ n + + φ ^ ∣ 2 d x = lim n → ∞ I ε n ( s w ^ n + + φ ^ n ) ⩽ lim n → ∞ I ε n ( m ε n ( w n ) ) = lim n → ∞ Φ ε n ( w n ) = c V 0 ,

which contradicts with the fact c V 0 < c V ∞ , where φ ^ n ( x ) = φ ^ ( x − y n ) for all n ∈ N .

If the case that ε n y n → y for some y ∈ R N does occur. Using a similar argument we can show that c V ( y ) ⩽ c V 0 and I V ( y ) ′ ( z ^ ) = 0 . On the other hand, if V 0 < V ( y ) , we deduce from Lemma 3.2 that c V 0 < c V ( y ) , which is a contradiction. Hence, according to ( V 1 ) we have V ( y ) = V 0 and y = x i for some i = 1 , … , k . Moreover, we can obtain

Q ε n ( w n ) = ∫ R N ξ ( ε x ) ∣ w n ∣ 2 d x ∫ R N ∣ w n ∣ 2 d x = ∫ R N ξ ( ε n ( x + y n ) ) ∣ w ^ n ∣ 2 d x ∫ R N ∣ w ^ n ∣ 2 d x → x i ∈ L ϱ 0 ∕ 2 .

This shows that Q ε n ( w n ) ∈ L ϱ 0 ∕ 2 for n large enough, which is absurd. We complete the proof.□

From now on, we will use the following notations:

S ε i ≔ { z ∈ S + : ∣ Q ε ( z ) − x i ∣ < ϱ 0 } and ∂ S ε i ≔ { z ∈ S + : ∣ Q ε ( z ) − x i ∣ = ϱ 0 } ,

and the numbers

c ε i = inf z ∈ S ε i Φ ε ( z ) and c ˜ ε i = inf z ∈ ∂ S ε i Φ ε ( z ) .

Lemma 4.5

There exists ε 0 > 0 such that

c ε i < c V 0 + π a n d c ε i < c ˜ ε i ,

for all ε ∈ ( 0 , ε 0 ) , where π = 1 2 ( c V ∞ − c V 0 ) > 0 .

Proof

Let z ∈ E be a ground state solution of limit system (4.1), then I V 0 ( z ) = c V 0 and I V 0 ′ ( z ) = 0 . Setting w = m V 0 − 1 ( w ) . Then by Lemma 2.9 we have

w ∈ S + , Φ V 0 ( w ) = c V 0 , and Φ V 0 ′ ( w ) = 0 .

For i ∈ { 1 , … , k } and ε > 0 , let us define the function w ˜ ε i = w ( ( ε x − x i ) ∕ ε ) . Evidently, w ˜ ε i ∈ S + . Letting z ε i = m ε ( w ˜ ε i ) . We prove the following conclusion holds

(4.8) limsup ε → 0 Φ ε ( w ˜ ε i ) = limsup ε → 0 I ε ( z ε i ) ⩽ c V 0 , ∀ i ∈ { 1 , … , k } .

Indeed, since z ε i = t ε ( w ˜ ε i ) + + φ ε for some t ε > 0 and φ ε ∈ E + , using a simple changing variable we can obtain

I ε ( z ε i ) = t ε 2 2 ‖ w + ‖ 2 − 1 2 ‖ φ ^ ε ‖ 2 + 1 2 ∫ R N V ( ε x + x i ) ∣ t ε w + φ ^ ε ∣ 2 d x − ∫ R N H ( ∣ t ε w + φ ^ ε ∣ ) d x .

Similarly, using the previous comments we can see that { z ε i } is bounded in E , and so, we may assume that t ε → t 0 and φ ^ ε → φ ^ as ε → 0 . Therefore, following the proof of Lemma 4.1, we obtain

limsup ε → 0 I ε ( z ε i ) ⩽ t 0 2 2 ‖ w + ‖ 2 − 1 2 ‖ φ ^ ‖ 2 + V 0 2 ∫ R N ∣ t 0 w + φ ^ ∣ 2 d x − ∫ R N H ( ∣ t 0 w + φ ^ ∣ ) d x ⩽ Φ V 0 ( w ) = c V 0

for all i ∈ { 1 , … , k } . Obviously, it follows that (4.8) holds.

Once Q ε ( w ˜ ε i ) → x i as ε → 0 , it shows that for ε small enough

w ˜ ε i ∈ S ε i and Φ ε ( w ˜ ε i ) < c V 0 + π 0 4

for some π 0 > 0 . Thus, there exists ε 1 > 0 such that

(4.9) c ε i < c V 0 + π 0 4 , for all ε ∈ ( 0 , ε 1 ) .

Then, decreasing π 0 if necessary,

c ε i < c V 0 + π , for all ε ∈ ( 0 , ε 1 ) ,

showing the first inequality.

In order to prove the second inequality, we recall that if z ∈ ∂ S ε i , then

z ∈ S + and ∣ Q ε ( z ) − x i ∣ > ϱ 0 2 ,

leading to Q ε ( z ) ∉ L ϱ 0 ∕ 2 . Therefore, using Lemma 4.4 we know that there exists ε 2 > 0 such that

Φ ε ( z ) > c V 0 + π 0 for all z ∈ ∂ S ε i and ε ∈ ( 0 , ε 2 ) ,

from where it follows that

(4.10) c ˜ ε i = inf z ∈ ∂ S ε i Φ ε ( z ) > c V 0 + π 0 , ∀ ε ∈ ( 0 , ε 2 ) .

Evidently, from (4.9) and (4.10) we can infer that

c ε i < c ˜ ε i , ∀ ε ∈ ( 0 , ε 0 ) ,

where ε 0 = min { ε 1 , ε 2 } . This ends the proof of lemma.□

Finally, we are going to prove Theorem 1.1.

Proof of Theorem 1.1 (completed)

According to Lemma 4.5, there exists ε 0 > 0 such that

c ε i < c ˜ ε i , ∀ ε ∈ ( 0 , ε 0 ) .

Arguing as in [1, Theorem 1.1], we can apply the Ekeland’s variational principle to obtain a ( P S ) c ε i sequence { z n i } ⊂ S ε i for Φ ε . Lemma 4.5 yields that c ε i < c V 0 + π , then by Lemma 4.3 there is z i ∈ E \ { 0 } such that z n i → z i in E . Therefore, we have

z i ∈ S ε i , Φ ε ( z i ) = c ε i , and Φ ε ′ ( z i ) = 0 .

Observe that, since

Q ε ( z i ) ∈ B ϱ 0 ( x i ) ¯ , Q ε ( z j ) ∈ B ϱ 0 ( x j ) ¯ , and B ϱ 0 ( x i ) ¯ ∩ B ϱ 0 ( x j ) ¯ = ∅ for i ≠ j ,

we derive that z i ≠ z j for i ≠ j with 1 ⩽ i , j ⩽ k . Thus, Φ ε possess at least k nontrivial critical points for all ε ∈ ( 0 , ε 0 ) on S + . Taking advantage of Lemma 2.9 we know that I ε possess at least k nontrivial critical points for all ε ∈ ( 0 , ε 0 ) on E . Going back to system (1.1) with the variable substitution x ↦ x ∕ ε , then we see that system (1.1) has at least k semiclassical solutions for each ε ∈ ( 0 , ε 0 ) , completing the proof of Theorem 1.1.□

Acknowledgements

This work was supported by the National Natural Science Foundation of China (12271152), the Natural Science Foundation of Hunan Province (2022JJ30200), and the Key project of Scientific Research Project of Department of Education of Hunan Province (22A0461, 23A0478).

  1. Conflict of interest: The authors declare that they have no competing interests.

  2. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2023-08-29
Revised: 2023-10-25
Accepted: 2024-01-26
Published Online: 2024-03-12

© 2024 the author(s), published by De Gruyter

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

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  39. Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
  40. Ground state solutions for magnetic Schrödinger equations with polynomial growth
  41. Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
  42. Dynamics for wave equations connected in parallel with nonlinear localized damping
  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
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  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
https://doi.org/10.1515/anona-2023-0139
Keywords for this article
Hamiltonian elliptic system; semiclassical solutions; multiplicity
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  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
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  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
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  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
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  77. Nonlocal heat equations with generalized fractional Laplacian
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  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
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  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
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