Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

Comparedwith some existing issues, themost interesting feature of this paper is thatwe assume that the nonlinearity satis es a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and in nitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.


Introduction and main results
We study the following nonlinear Hamiltonian elliptic systems with gradient term −∆u + b(x) · ∇u + V(x)u = Hv (x, u, v) and H ∈ C (R N × R , R). In the present paper, our main goal is to establish some new existence results of ground state solutions and in nitely many geometrically distinct solutions of system ( . ) under some suitable conditions on the potential V and the nonlinearity H. This type of systems arises when one is looking for the standing wave solutions to system of di usion equations which comes from the time-space di usion processes and is related to the Schrödinger equations. It appears in various elds, such as physics and chemistry, quantum mechanics, nance, dynamic programming, optimization and control theory and Brownian motions. For more details in the application backgrounds, we refer the readers to see the monographs [13] and [19].
In recent years, the Hamiltonian elliptic system has being extensively investigated in the literatures based on various assumptions on the potential and nonlinearity. But most of them focused on the case b = , namely, −∆u + V(x)u = Hv (x, u, v) in For instance, the papers [2,4,7,9,10,23,31,35] studied the super-quadratic growth case, and the asymptotically quadratic case can be found in [18,29,34]. Moreover, the existence of nontrivial solutions, ground state solutions, multiple solutions and semiclassical solutions were obtained in these works by using various variational arguments, such as dual methods, reduction methods, generalized mountain pass theorem, generalized linking theorem and many others. For further related topics including the Hamiltonian systems, we refer the reader to [3,11,12,21] and their references. When b ≠ , as we all know, there are a few works devoted to the existence and multiplicity of solutions of system (1.1), see [15,30,32,36,40]. For this case, since the appearance of the gradient term in system itself, system (1.1) has some di erences and di culties comparing to system (1.2). For example, the variational framework for the case b = cannot work any longer in this case, then the rst problem is how to establish a suitable variational framework. To solve this problem, Zhao and Ding [32] handled ( . ) as a generalized Hamiltonian system, and established a strongly inde nite variational framework by studying the structure of essential spectrum of Hamiltonian operator. In this framework, the existence and multiplicity of solutions were obtained by using critical point theorems of strongly inde nite functional and reduction method for system (1.1) with periodic and non-periodic asymptotically quadratic growth condition. After that, Zhang et al. [36] studied the periodic super-quadratic case and proved the existence of ground state solutions by means of the linking and concentration compactness arguments. Later, this result has been extended to more general nonlinearity model by Liao et al. [15]. An asymptotically periodic case was considered in [40], and some properties of ground state solutions were obtained by constructing linking levels and analyzing behavior of Cerami sequence. The existence of least energy solution for the non-periodic super-quadratic case was studied in [30]. In [37], the authors studied the Hamiltonian elliptic system with inverse square potential of the form u, v) in R N , and the ground state solutions was obtained by using non-Nehari manifold developed by Tang [25]. Moreover, some asymptotic behaviors of ground state solutions, such as the monotonicity and convergence property of ground state energy, were also explored as parameter µ tends to . In addition, the singularly perturbed problem has been considered in [38,39]. More precisely, the authors proved the existence of semi-classical ground state solutions, and shown some new concentration phenomena of semi-classical states.
It is worth pointing out that, for the aforementioned papers about super-quadratic problems, the classical condition frequently used in the literature is due to Ambrosetti and Rabinowitz [1] (AR)there exists θ > such that, for each x ∈ R N , z ∈ R \{ }, there holds It is well known that condition (AR) has been used in a technical but crucial way not only in establishing the geometry structure of the energy functional but also in proving the boundedness of Palais-Smale sequences. Via a straightforward calculation, we can see easily that H(x, z) ≥ c|z| θ for large values of |z| under condition (AR). Clearly, it puts strict constrains on the growth at in nity, and therefore it is natural to consider a weaker condition. After that, there are many works devoted to replacing condition (AR) with a more natural superquadratic condition Condition (SQ) was rst introduced by Liu and Wang [16] in studying the superlinear problem of the elliptic equation. Moreover, it plays a crucial role in verifying the link geometry and in showing the boundedness of Cerami sequences for the energy functional. Indeed, condition (SQ) is essential to prove the existence of nontrivial solutions in all literature.
Recently, motivated by [27] where the authors introduced a local version of super-quadratic condition when studying the scalar eld Schrödinger equation, Zhang and Liao [33] obtained a new existence result of nontrivial solution for system ( . ) under the following condition Here condition (f ) is called local super-quadratic condition which allows the nonlinearity to be superquadratic at some domains and asymptotically quadratic at other domains. Hence it weakens the usual global super-quadratic condition (SQ). However, to the best of our knowledge, it seems that ground state solutions and multiplicity results for system ( . ) with local super-quadratic condition (f ) have not been studied so far. As a complement, in this paper we will continue the work in [33] in this direction. More precisely, our purpose in this paper is twofold, one is to prove the existence of ground states, that is, the least energy nontrivial solutions; the other is to establish the existence of in nitely many geometrically distinct solutions.
In what follows, in order to state our statements we assume that the following conditions: H(x, z) ≥ , and there exist c > , δ ∈ ( , a) and σ ∈ ( , ) such that For the sake of convenience to describe our results, here we rst need to give some notations. Let E be the Hilbert space with an orthogonal decomposition E = E − ⊕ E + , and let Φ denote the energy functional of system ( . ), where E and Φ will be de ned in Section . We de ne the generalized Nehari manifold and the critical points set N := {z ∈ E \ { } : Φ (z) = } of Φ. According to [20,24], the set M is a natural constraint and it contains all nontrivial critical points of Φ. Obviously, N ⊂ M . We say that a nontrivial solution z is ground state solution if its energy attians the minimum among all nontrivial critical points. Additionally, observe that, due to the periodicity of b, V and H, if z is a solution of system ( . ), then so is k * z for all k ∈ Z N , where (k * z)(x) = z(x + k). Two solutions z and z are said to be geometrically distinct if k * z ≠ z for all k ∈ Z N .
We are now in position to state the main results of this paper. On the existence of ground state solutions we have the following results. On multiplicity results of solutions we have the following theorems. As observed in [33], the rst author and co-author only proved the existence result of nontrivial solution by using generalized linking theorem under the conditions of Theorem . , and the other related results are all unknown. Compared to the result in [33], the results obtained in this paper can be viewed as a continution of [33], and seem more delicate.
To prove Theorem . and Theorem . , some arguments are in order. Here we rst introduce the problem of ground state solution. For the Theorem . , based on the result in [33], we can see that N ≠ ∅. By using minimization method and concentration compactness argument, the conclusion for ground state solution in Theorem . holds. However, Theorem . seems more complicated than Theorem . , and there are many new di culties. Generally speaking, M contains in nitely many elements of E, while N may contain only one element. So it becomes more di cult to nd a ground state solutionz which satis es Φ(z) = inf M Φ than one that satis es Φ(z) = inf N Φ. Additionally, we note that the variational structure of system ( . ) is strongly inde nite, thus the usual Nehari manifold method cannot be applied directly. To overcome these di culties, in the spirit of [20,24] we choose the generalized Nehari manifold M to work. More precisely, we intend to make use of the non-Nehari manifold method developed by Tang [25] to complete the proof of Theorem . . The main idea of this method is to construct a minimizing Cerami sequence for energy functional Φ outside M by using the diagonal method and linking argument.
We would like to point out that the global super-quadratic (SQ) is indispensable in verifying the linking geometry structure and constructing Cerami sequence by the diagonal method and linking argument, see [37,40]. Unfortunately, in the present paper we have no global information on the nonlinearity like (SQ), then the non-Nehari manifold method seems not work to our problem under the local super-quaratic condition. So, some new methods and techniques need to be introduced. Motivated by [26], we present a perturbation approach by adding a perturbation term of power function. More precisely, for µ ∈ ( , ] and p ∈ ( , * ), we consider the following perturbation problem and its associated functional is as follows In such a way, the modi ed nonlinearity satis es the global super-quadratic condition (SQ), then, by using the non-Nehari method in [25] and concentration compactness principle, we can obtain a ground state solution zµ of the perturbation problem. Finally, by passing to the limit and by some special techniques, we show the convergence as µ → of {zn} towards to a ground state solution of the original problem. For the sake of completeness, next we consider the multiplicity results of system ( . ). In view of Theorem . and Theorem . , we know that N ≠ ∅. To prove the existence of in nitely many geometrically distinct solutions, we choose a subset F of N such that F = −F and each orbit O(w) ⊂ N has a unique representative in F due to Φ(z) = Φ(−z), and then show that the set F is in nite. To do this, inspired by [24,26], using some arguments about deformation type and Krasnoselskii genus, we nd in nitely many geometrically distinct solutions.
The remainder of this paper is organized as follows. In Section , we introduce the variational setting of the problem and present some useful preliminaries. In Section , we prove that the perturbation problem has a ground state solution. In Section , we give the proof of the existence of ground state solutions in Theorem . and Theorem . , respectively. At last, the existence of in nitely many geometrically distinct solutions is established in Section .

Variational setting and preliminaries
Below by | · |q we denote the usual L q -norm, (·, ·) denotes the usual L inner product, c, c i or C i stand for di erent positive constants. For the sake of convenience, we need the following notations. Let then system ( . ) can be rewritten as Az = Hz(x, z).
According to [8], in this way, system ( . ) can be regarded as a Hamiltonian system. Denote by σ(A) and σe(A) the spectrum and the essential spectrum of operator A, respectively. In order to establish a suitable variational framework for system ( . ), we need to utilize some properties of spectrum of operator A due to [32]. Observe that, it follows from Lemma . and . that the space L possesses the following orthogonal decomposition such that A is negative de nite (resp. positive de nite) in L − (resp. L + ). Let |A| denote the absolute value of A and |A| / be the square root of |A|. Let E = D(|A| / ) be the Hilbert space with the inner product (z, w) = (|A| / z, |A| / w) and norm z = (z, z) / . Moreover, it is obvious that E possesses the following decomposition which is orthogonal with respect to the inner products (·, ·) and (·, ·). Note that E = H := H (R N , R ) and · is equivalent to the usual norm of H (see [32]). Then E embeds continuously into L q for all q ∈ [ , * ] and compactly into L q loc for all q ∈ [ , * ). Moreover, there exists a positive constant πq > such that for all (2.1) On the one hand, by virtue of (f ) and (f ), for any ϵ > , there exists a positive constant cϵ such that On the other hand, it follows from (f ) that In fact, given z ≠ , (f ) implies that This, together with the monotonicity of h(x, |z|), implies that On E we de ne the energy functional Φ corresponding to system (1.1) as follows By the above facts and some standard arguments, we can easily see that Φ ∈ C (E, R) and the critical points of Φ are solutions of system ( . ) (see [6,28]), and for z, Now we discuss the linking geometry structure of the energy functional Φ.
The proof of Lemma . is standard, and the details can be seen in [36] and hence is omitted.
Without loss of generality, we can assume that Ω ⊂ R N is a bounded domain. We chooseẽ ∈ C ∞ (R N ) ∩ C ∞ (Ω) such that ẽ + − ẽ − = (Aẽ,ẽ) ≥ , which implies thatẽ + ≠ . Based on this fact, using the special technique as in [33], we can obtain the following lemma, which is very critical in our arguments, the proof can be found in [33].
We recall that a functional Φ ∈ C (E, R) is said to be weakly sequentially lower semi-continuous if for any un u in E one has Φ(u) ≤ lim inf n→∞ Φ(un), and Φ is said to be weakly sequentially continuous if We say that Φ satisfy the (C)c-condition if any (C)c-sequence has a convergent subsequence.
To prove the main results, we need the following generalized linking theorem in [17].
Lemma 2.6. Let X be a real Hilbert space with X = X − ⊕ X + , and let Φ ∈ C (X, R) be of the form Suppose that the following assumptions are satis ed:

bounded from below and weakly sequentially lower semi-continuous; (A )Ψ is weakly sequentially continuous;
(A )there exist R > ρ > and e ∈ X + with e = such that

The perturbation problem
In this section, we will in the sequel focus on the perturbation problem ( . ) and study the existence of ground state solution. We de ne the perturbation functional Φµ of Φ If mµ is attained by zµ ∈ Mµ, then zµ is a critical point of Φµ. Since mµ is the lowest level for Φµ, then zµ is called a ground state solution of the perturbation problem ( .
Since gµ(x, s) is increasing in s on ( , +∞) due to (f ), we can obtain Gµ(x, t) ≤ for t ≥ by using some arguments in [37,40]. So, we get the rst conclusion from the above formula. If z ∈ Mµ, then Φ µ (z)z = Φ µ (z)w = , then the second conclusion holds.
For convenience of notation, we write E(z) := E − ⊕ R + z = E − ⊕ R + z + for z ∈ E\E − . Let z ∈ Mµ, then Lemma 3.2 implies that z is the global maximum of Φµ| E(z) . Next we shall verify that Φµ possesses the linking structure. Proof. (i) For z ∈ E + and µ ∈ ( , ], by ( . ) and (2.2), we obtain It is easy to see that there exist positive constants ρ and α both independent of µ such that inf Sρ Φ ≥ α due to the arbitrariness of ϵ > . So the second inequality holds. Note that for every z ∈ Mµ there exists s > such that sz + ∈ E(z) ∩ Sρ. Hence, by Lemma 3.2 we know that the rst inequality holds.

Lemma 3.4.
Suppose that (f )-(f ) are satis ed. Then for any e ∈ E + , sup Φµ(E − ⊕ R + e) < ∞, and there is Re > such that In particular, there is a R > ρ such that sup Φµ(∂Q R ) ≤ for R ≥ R , where Proof. Since the modi ed nonlinearity Gµ(x, z) satis es the global super-quadratic condition (SQ), the proof is standard, see [36,40]. So we omit it here.
Employing Lemmas . , . , . and . , we have To prove the existence of ground state solutions for the perturbation problem (1.3), next we construct a special Cerami sequence by using diagonal method (see [25]), which is very important in our arguments. Proof. Choose ξ k ∈ Mµ such that By virtue of Lemma 3.3, ξ + k ≥ mµ > . Set e k = ξ + k / ξ + k . Then e k ∈ E + and e k = . From Lemma 3.4, it follows that there exists R k such that sup Φµ(∂Q k ) ≤ , where Hence, using Lemma 3.5 to the above set Q k , there exist a constant c µ,k ∈ [κ, sup Φµ(Q k )] and a sequence {z k,n } n∈N ⊂ E satisfying On the other hand, by Lemma 3.2, one can get that Since ξ k ∈ Q k , it follows from (3.4)) and (3.6) that Φµ(ξ k ) = sup Φµ(Q k ). Hence, by (3.
We can choose a sequence {n k } ⊂ N such that Φµ(z k,n k ) < mµ + k and Φ µ (z k,n k ) ( + z k,n k ) < k , k ∈ N.
Let z k = z k,n k , k ∈ N. Then, up to a subsequence, we have Similar to the proof of Lemma . , we have the following result. If δ = , by Lions' concentration compactness principle (see [14,28]), then w + n → in L q for any < q < * . It follows from ( . ) that for any s > , Observe that, from ( . ) and ( . ), we deduce that Let tn = s/ zn , then by ( . ) we have On the other hand, by ( . ) we get Φ µ (zn), zn ≤ z + n − z − n , which implies that This shows that w + n ≥ c for some c > . Hence, ( . ) yields a contradiction if s is large enough. Then δ > . Up to a subsequence, we assume kn ∈ Z N such that Letwn(x) = wn(x + kn). By the periodicity of b and V, we have that wn = w n = and Therefore, passing to a subsequence,w + n →w + in L loc andw + ≠ . Note that ifw ≠ , then |zn(x + kn)| = |wn(x)| zn → ∞. Since Gµ satis es condition (SQ), then it follows from Fatou's lemma that which is a contradiction. So, {zn} is bounded in E.
The proofs of Theorems . and .
In this section, we give the proofs of Theorem . and Theorem . . Indeed, for Theorem . , we will make use of the conclusion of the perturbation problem ( . ), by passing to the limit as µ → and some special techniques, to show that zµ towards to a ground state solution z of the original problem. For Theorem . , we will use a minimization method and concentration compactness argument to complete the proof.
Before proving Theorem . , we need to show that the following result holds. Moreover, this result will be used in the proof of Theorem . and Theorem . .
Next, we show that conclusion (2) holds under the assumptions of Theorem . . Indeed, suppose to the contrary that there exists a sequence {zn} ⊂ N such that Φ(zn) → and Φ (zn) = . Clearly, {zn} is a (C)sequence of Φ. Moreover, by some arguments as in [33], {zn} is bounded in E. Together with conclusion (1), we know that α ≤ zn ≤ C for some C > . Observe that (4.9) Let wn = zn / zn , then wn = . Set According to Lemma . , there holds a|wn| ≤ wn . Using this fact, we get On the other hand, using the fact α ≤ zn ≤ C, (f ), ( . ) and Hölder inequality we obtain (4.11) Therefore, it follows from ( . ) and ( . ) that which is a contradiction since δ ∈ ( , a). This shows that conclusion (2) holds. The proofs of Theorems . and .
In this section, we are devoted to looking for in nitely many geometrically distinct solutions for system ( . ), and give the proofs of Theorem . and Theorem . . To this end, we need some notations. For d ≥ e > −∞ we put Using some standard arguments in [6,26], we can obtain the following results without proof.
In the following we discuss further the (C)c-sequence. Let [l] denote the integer part of l ∈ R. Combining Lemma . , Lemma . and some standard arguments, we have the following lemma (see Coti-Zelati and Rabinowitz [5]). As in [5,24], we choose a subset F of N such that F = −F and each orbit O(w) ⊂ N has a unique representative in F . In order to prove Theorems . and . , it su ces to show that the set F is in nite. Arguing by contradiction, we assume that F is a nite set.
For any c ≥ c , as in [5], let Following some arguments in [5] and [24], we have The following lemma plays an important role in expressing the discreteness property of the Cerami sequence. then {zn} is a (C)c-sequence. By Lemma . , up to a subsequence, there exists {wn} ⊂ Fc such that zn − wn → . Moreover, there exists un ∈ S such that zn − un ≤ εn for n ∈ N. Since wn ∈ Fc and un ∈ S, from the above facts, we deduce that < δ ≤ un − wn ≤ zn − un + zn − wn ≤ εn + o( ) = o( ).
This contradiction shows that ( . ) holds. Since Φ is even, using the deformation lemma from [28], there exists an odd and continuous function η : η(t, z)) is nonincreasing for all z ∈ E.
Let ϕ(z) = η( , z), hence it follows from the above conclusions that ϕ has the asserted properties.
Proof. Since F is a nite set and symmetric, then we can assume that Fc = {±zn : n ∈ N}. Plainly, Fc is also nite and symmetric. Moreover, by [22,Example 7.2], we know γ(Fc) = . Additionlly, since δ ∈ ( , γc / ), then U δ (Fc) is a closed and symmetric set. Therefore, it follows from the continuity property of the genus that γ(U δ (Fc)) = .
Proof of Theorems . and . . For j ∈ N, we consider the family Σ j of all closed and symmetric subsets A ⊂ E\{ } with γ(A ) ≥ j. Moreover, we consider the nondecreasing sequence of Lusternik-Schnirelman values for Φ de ned by c k := inf{c ≥ c : γ(Φ c ) ≥ k}, k ∈ N.
Then ζ is odd and continuous. Hence, by the mapping property of the genus, we have γ(Φ c k +ε ) ≤ γ(Φ c k −ε ). Obviously, this contradicts with the de nition of c k . Next, we show c k < c k+ for all k ∈ N. Indeed, for each k, from Lemma . , it follows that there exist δ k ∈ ( , γc k / ), ε k > and an odd and continuous map ϕ : Φ c k +ε k \ U δ k (Fc k ) → Φ c k −ε k . Again by the mapping property of the genus, we have γ(Φ c k +ε k \ U δ k (Fc k )) ≤ γ(Φ c k −ε k ) ≤ k − . Moreover, by the subadditivity of the genus and Lemma . , we obtain k ≤ γ(Φ c k +ε k ) ≤ γ(U δ k (Fc k )) + k − = k, which implies that γ(Φ c k +ε k ) = k. On the other hand, note that c k+ ≥ c k . If c k+ = c k , then γ(Φ c k +ε k ) ≥ k + , a contradiction. Hence, c k+ > c k .
Finally, it follows from ( . ) that there is an in nite sequence {z k } of pairs of geometrically distinct critical points of Φ with Φ(z k ) = c k , contrary to ( . ). The proof is nished.