New asymptotically quadratic conditions for Hamiltonian elliptic systems

: This paper is concerned with the following Hamiltonian elliptic system where z = ( u , v ) : R N → R 2 , V ( x ) and W ( x , z ) are 1-periodic in x . By making use of variational approach for strongly indefinite problems, we obtain a new existence result of nontrivial solution under new conditions that the nonlinearity W ( x , z ) := 12 V ∞ ( x ) | Az | 2 + F ( x , z ) is general asymptotically quadratic, where V ∞ ( x ) ∈ ( R N , R ) is 1-periodic in x and inf R N V ∞ ( x ) > min R N V ( x ), and A is a symmetric non-negative definite matrix.

In the past few decades, by using variational techniques, a number of important results of the existence and multiplicity of solutions for system (1.2) defined on the bounded domain Ω have been established with W satisfying various conditions, see for example [7,9,10]. Recently, many authors began to focus on system (1.2) defined on the whole space R N . Their most interesting studies were to establish the existence of multiple solutions, ground states and semiclassical states, see [1, 2, 4, 6, 12, 18, 21, 25-29, 31, 32] and their references therein. In these works, a huge machinery is needed to obtain existence and multiplicity of solutions, such as fractional Sobolev spaces, generalized mountain pass theorem, generalized linking theorem, reduction Nehari method and many others. Besides, based on variational arguments, some related problems involving the nonlocal elliptic equations have been received increasingly more attention on mathematical studies . Tang and Chen [20] studied the ground state solution of Nehari-Pohozaev type for the nonlocal Schrödinger-Kirchhoff problems by developing some new analytic techniques. More relevant results and recent developments, we refer the readers to [16] for elliptic problems and the monograph [17] for nonlocal fractional problems.
One of the main difficulties in dealing with system (1.2) relies on the lack of embedding compactness due to the unboundedness of the domain. In some of the above quoted papers this difficulty was overcame by imposing periodicity condition both on the potential V and the nonlinearity W. Along this direction, the papers [8,13,14,32] studied system (1.2) with periodic and global super-quadratic growth, and the existence and multiplicity results are obtained. Subsequently, the authors in [30] weakened the global super-quadratic case to the local super-quadratic case and proved the existence of ground state solutions and infinitely many geometrically distinct solutions by using a new perturbation approach developed by Tang and his collaborators [22,23].
Motivated by the researches about the Hamiltonian elliptic systems, we continue to study system (1.2) under general conditions, and assume the following basic assumptions.
In the aforementioned references [12,[25][26][27][28], the following asymptotically quadratic condition and other technique conditions for the nonlinearity W are commonly assumed: (W5) W(x, z) := 1 2 Wz(x, z)z − W(x, z) ≥ 0, and there exist δ 0 > 0 such that Observe that conditions (W4) and (W5) play an important role for showing that any Palais-Smale sequence or Cerami sequence is bounded in the aforementioned works. However, there are many functions do not satisfy these conditions, for example, .
In a recent paper [14], making use of some special techniques, Liao, Tang, Zhang and Qin studied the existence of solutions for system (1.2) under more general super-quadratic conditions, that is, Clearly, this condition is weaker than the usual super-quadratic condition Very recently, the singularly perturbed problem with super-quadratic condition (SQ) has been investigated in [21], where the authors proved the existence of semiclassical ground state solutions and generalized the results in [4].
Inspired by super-quadratic case [14] and [21], we further consider the general periodic asymptotically quadratic case and establish the existence result of solutions. In addition to (V), (W1) and (W2), we introduce the following new asymptotically quadratic conditions for the nonlinearity W: (W3 ′ ) there exists symmetric non-negative definite matrix It is worth pointing out that conditions (W3 ′ ), (W4 ′ ) and (W5 ′ ) are different from usual conditions (W3), (W4) and (W5) and weaken these conditions. To the best of our knowledge, it seems that there is no work considered this problem in the literature before. So this result obtained in this paper is new, moreover, it can be viewed as a complement and an extension of [14] and [21]. However, it is difficult for us to prove the linking geometry and boundedness of Cerami sequences under the conditions (W3 ′ ), (W4 ′ ) and (W5 ′ ) since the arguments as [26,27] (depending on the behavior of W(x, z) as |z| 2 = |u| 2 + |v| 2 → ∞) cannot be applied directly. To do this, some new techniques need to be introduced in the proof.
Based on the conditions given above on V and W, we can get the following theorem. Before proceeding to the proof of Theorem 1.1, we give a nonlinear example to illustrate the assumptions.

Example 1.2. For example, let
By a straightforward computation, we can see that all conditions (W1), (W2), (W3 ′ ), (W4 ′ ) and (W5 ′ ) are satisfied with a = 5 2 and The remainder of this paper is organized as follows. In Sect. 2, we introduce the variational setting of system (1.1). In Sect. 3, we analyze the geometry structure of the functional and property of Cerami sequence, and give the proof of Theorem 1.1.

Variational setting
Throughout this paper, we make use of the following notations. · s denotes the usual norm of the space L s , 1 ≤ s ≤ ∞; (·, ·) 2 denotes the usual L 2 inner product; c or c i (i = 1, 2, . . . ) are some different positive constants.
In the following, we establish the variational setting of system (1.2). According to condition (V), we define the following Hilbert space and the reduced norm Let the working space E = H × H. Then E is a Hilbert space with the standard inner product , and the corresponding norm Observe that, the natural functional associated with system (1.2) is given by Moreover, according to conditions (V), (W1) and (W2), it is easy to prove that Φ ∈ C 1 (E, R), and for any z = (u, v) and η = (φ, ψ) ∈ E, there holds Following the idea of De Figueiredo and Felmer [7] (see also Hulshof and Van Der Vorst [9]), for any z = (u, v) and w = (w 1 , w 2 ) ∈ E, we introduce a bilinear form on E × E as It is clear that B[z, w] is continuous and symmetric, and hence B induces a self-adjoint bounded linear op- By a direct computation, we can deduce that Moreover, it is easy to see that 1 and −1 are two eigenvalues of the operator L , and the corresponding eigenspaces are Hence, based on the above fact, the working space E has the following decomposition Then we have Now we define the functional F : E → R as Computing directly, we get Therefore, the functional Φ defined by (2.1) can be rewritten the following form Obviously, Φ is strongly indefinite and the critical points of Φ are solutions of system (1.2) (see [3]), and for z, φ ∈ E we have On the other hand, according to the embedding theorem and condition (V), H embeds continuously into L p (R N ) for all p ∈ [2, 2 * ] and compactly into L p loc (R N ) for all p ∈ [1, 2 * ). Therefore, it is easy to see that E embeds continuously into L p := L p (R N ) × L p (R N ) for all p ∈ [2, 2 * ] and compactly into L p loc := L p loc (R N ) × L p loc (R N ) for all p ∈ [1, 2 * ).

Proof of main result
In this section, we will in the sequel focus on the proof of Theorem 1.1. Firstly, we need verify the linking geometry structure of the functional Φ.
Combining the above fact, and letting n → ∞ in (3.16) we have (ξ + −ξ − , η) − (V ∞ (x)Aξ , η) 2 = 0, (3.17) for each η = (μ, ν) ∈ C ∞ 0 (R N ) × C ∞ 0 (R N ), which implies that Hence, from (3.17) we can duduce that (φ, ψ) is an eigenfunction of J := A − V∞(x)A, which contradicts with the fact that J has only continuous spectrum since the matrix A is symmetric non-negative, V(x) and V ∞(x) are 1-periodic in x. Therefore, this shows that {zn} is bounded in E. Finally, using a standard translation argument, we can see that {zn} converges weakly (up to a subsequence) to some z 0 ≠ 0, and Φ ′ (z 0 ) = 0. This shows z 0 is a nontrivial solution of system (1.2). The proof of Theorem is completed.