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

: This article is concerned with the following Hamiltonian elliptic system:


Introduction and main result
In this article, we deal with a class of singularly perturbed Hamiltonian elliptic system with gradient term where = → z u v , : N 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: 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][16][17]21,27,[29][30][31][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 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 where Ω is a bounded domain in 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 nonli- nearity 1 and min lim ; , and for some > c 0 0 and ∈ p 2, 2* ( ), there holds We state in what follows the main result of this article.
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.
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 N ( ) for ⩽ ⩽ +∞ s 1 ; • ⋅ ⋅ , 2 ( ) denotes the inner product of L N 2 ( ); • c, c i , C i denote (possibly different) any positive constants, whose values are relevant; • σ A ( ) and σ A e ( ) 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 εx, we can rewrite system (1.1) as the following equivalent system: Evidently, we can see that if ) is a solution of system (2.1), then 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 Then system (2.1) can be rewritten as 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 Lemma 2.2.We have the following two conclusions about the spectrum of A: ) 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: Since ⊂ E L 2 , it follows that E has the following decomposition: which is orthogonal with respect to the inner products ⋅ ⋅ , 2 ( ) and ⋅ ⋅ , ( ).Moreover, using the polar decomposition of A we can obtain that where ∈ z E and ⋅ denotes the usual inner product in 2 .Employing the polar decomposition of A, the energy functional I ε has another representation as follows: From Lemma 2.2 we can see that I ε is strongly indefinite.Our hypotheses imply that ∈ I C E, ε 1 ( ).By standard argument we know that critical points of I ε are solutions of system (2.1), and for ∈ z ψ E , , there holds We note that if ∈ z E is a nontrivial critical point of I ε , then we can obtain from (2.6) that On the other hand, for any 3), and (2.6) we obtain Therefore, we can see that all nontrivial critical points of I ε are in the space , 0 and , 0, , deeply studied by Szulkin and Weth [25].Following the terminology of Szulkin and Weth [25], the set ε N 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 , 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 and the convex subset We introduce a crucial estimate, which plays a key role in the method of the generalized Nehari manifold.
, and ⩾ t 0 with ≠ + z tz w, then we have the following estimate: .
In particular, let ∈ z ε N , ∈ − w E , and ⩾ t 0 with ≠ + z tz w, there holds On the one hand, using V 1 ( ) and (2.3) we deduce that 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 Then we have (a) there exist two positive constants ϱ and α such that , where Proof.(a) Let ∈ + z E , then from (2.3), (2.4), and (2.5) we infer that and > p 2, we can find that there exist two positive constants ρ and α both independent of ε such that , so from Lemma 2.3 we can conclude that Therefore, we show that conclusion (a) holds.
), and conclusion (a), we can obtain Thereby, it follows that Proof.Without loss of generality, we can assume that = z 1 ‖ ‖ for all ∈ z .Suppose by contradiction that there exist 3), and (2.6) we can derive that which yields that > s 0, and so = + ≠ − u sz u 0. If this is not true, then = s 0. From the above inequality we can see that in Ω 0 .Thereby, using f 3 ( ) and Fatou's lemma we obtain 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: and Evidently, (2.7) and (2.8) yield that the norm where the symbol ~denotes the equivalence of two norms.
Lemma 2.6.For each In other words, there exist unique > t 0 Proof.We follow some ideas found in [25].Indeed, according to Lemma 2.3, it suffices to show that ( ), we may assume that ∈ + z E .From Lemma 2.5, we find that there exists Next we need to show that I ε is weakly upper semicontinuous on Hence, using (2.8), Fatou's lemma and the weak lower semicontinuity of norm we can infer that where we use the following fact: 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 Proof.Suppose by contradiction that there is a sequence On account of Lions' concentration compactness principle, in the following we will discuss two cases: vanishing or nonvanishing.
).Therefore, for any > s 0, we deduce from (2.5) that  ( ) for ⩾ s 0, then using Lemma 2.3, (2.11), and (2.12) we can obtain which is absurd if s is large enough.So the vanishing case does not occur.
Let us consider the sequence ) and Fatou's lemma we can derive that 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 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: for a sequence \ 0 , then .
Without loss of generality, we may assume that . By Lemma 2.5, we find that there exists > R 0 such that  ( ) is bounded by Lemma 2.7.Up to a subsequence, we can assume that where > t 0 by Lemma 2.4.Furthermore, using Lemma 2.6 we can infer that Applying Fatou's lemma and the weak lower semicontinuity of norm, and combining (2.9) we can conclude that Moreover, the argument above also yields that . Therefore, we can see that 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 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 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: ) and for each ∈ + z S and ∈ is a critical point of I ε .Moreover, the corresponding values of Φ ε and I ε coincide and 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.
), we start by considering the autonomous problem: It is well known that the solutions of problem (3.1) are precisely critical points of the energy functional defined by Similar to the previous section, we define the associated generalized Nehari manifold , 0 and , 0, and the ground state energy value Employing the same arguments used in the previous section, we can know that for every  ( ) 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 Accordingly, we can define the maps Meanwhile, we consider the reduction functional 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 , κ N , 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 We now state the existence result of ground state solution for the autonomous problem (3.1).( ) , then from Lemma 2.9 we can see that ) 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 . In fact, using Ekeland's variational principle [26], there exists a sequence Using the fact that I κ and κ N are invariant under translations, we may assume that then, Lions' concentration compactness principle [26,Lemma 2.1] yields that → z 0 n in L q for any ∈ q 2, 2* ( ).From (2.5) we can conclude that and it follows that . The reverse inequality follows from the definition of c κ since ∈ ∼ z κ N .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 .
Proof.In what follows, let z κ1 and z κ2 be as ground state solution of I κ1 and I κ2 .Assume that . First of all, we prove that the function ↦ κ c κ is increasing.By Lemma 2.6, there exist > t 0 , it is easy to see that 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: 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 0 Computing directly, we have According to the monotonicity of c κ , we immediately obtain which together with (3.5), yields that On the other hand, since ⩽ c c κ κ n for all ∈ n , then we show that Indeed, let z n be the ground state solution of problem (3.1) with = κ κ n , then there exist > s 0 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 N { } such that for each ∈ n , we have If not, using Lions' concentration compactness principle we have Then we can check that Thereby, it follows that { }, hence 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 , we obtain Let us define , we have 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 ′ Following some ideas of proof [11,  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: For convenience, next we will use the notations I V 0 ( ) , c V 0 ( ) , and V 0 N ( ) to denote the associated energy func- tional, 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 Evidently, we can obtain and we complete the proof.□ As a byproduct of Lemma 4.1, we can directly obtain the following result.
. Next we establish compactness criteria for the functional Φ ε , which is crucial in our approach., where Proof , then similar to the proof of [34, Lemma 5.2], we obtain the following results: Observe that since ′ → I z ψ , 0 . Moreover, using (2.6) we can deduce that  for n large enough, which is absurd.We complete the proof.□ From now on, we will use the following notations: Similarly, using the previous comments we can see that z ε i { } is bounded in E, and so, we may assume that → t t ε 0 and → φ φ ε   as → ε 0. Therefore, following the proof of Lemma 4. , for all 0, .
Then, decreasing π 0 if necessary, showing the first inequality.
In order to prove the second inequality, we recall that if ∈ ∂ z ε i , then . Therefore, using Lemma 4.4 we know that there exists > ε 0 2 such that for all and 0, , from where it follows that Evidently, from (4.9) and (4., and Φ 0.
Observe that, since we derive that ≠ z z i j for ≠ i j with ⩽ ⩽ i j k 1 , .Thus, Φ ε possess at least k nontrivial critical points for all ∈ ε ε 0, 0 ( ) on + S .Taking advantage of Lemma 2.9 we know that I ε possess at least k nontrivial critical points for all ∈ ε ε 0, 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 ε 0, 0 ( ), completing the proof of Theorem 1.1.□ Hamiltonian elliptic system  21

2
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 A 1 2 D(| | ) be the domain of the self-adjoint operator ∕ A 1 2 | | , which is a Hilbert space equipped with the inner product we consider the following set introduced by Pankov[20]

.
Proof.Let ∈ z E be a ground state solution of limit system (4.1), then = 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 > Arguing as in [1, Theorem 1.1], we can apply the Ekeland's variational principle to obtain a PS c ε i