Multiplicity of Solutions for a Class of Noncooperative Elliptic Systems

In this paper, we consider the following noncooperative elliptic systems

, , in , , , in , 0 on , where Ω is a bounded domain in N  R with smooth boundary ∂Ω , , , λ δ γ are real parameters, and ( ) ∈ Ω ×   .We assume that F is subquadratic at zero with respect to the variables , u v .By using a variant Clark's theorem, we obtain infinitely many nontrivial solutions ( )

Introduction
In this paper, we consider the existence of nontrivial solutions for the following variational noncooperative elliptic system ( ) ( ) , , in , , , in , 0 on , where Ω is a bounded smooth domain in N  R with smooth boundary ∂Ω , the numbers , , λ δ γ are real parameters, and ( ) . Here F ∇ denotes the gradient of F in the variables u and v.
System (P) is the so-called noncooperative elliptic system, arising naturally a steady state in reaction-diffusion process that appears in chemical and biological phenomena, including the steady and unsteady state situation (see [1] [2] [3] [4] and references therein).System (P) has been extensively studied in last three decades by using the variational methods under various conditions (see [1]- [17]).In [1], Costa and Magalhães established the variational structure to the noncooperative elliptic systems and obtained several existence results under the nonquadraticity at infinity by using minimax methods.In [5], by using a variant fountain theorem, Bartsch and Figueiredo obtained infinitely many nontrivial solutions for nonlinear noncooperative elliptic systems when ( ) , , F x u v satis- fied the non-quadraticity condition at infinity.In [6], Guo obtained the existence and multiplicity of nontrivial solutions for resonant noncooperative elliptic systems by using a new version of Morse theory for strongly indefinite functionals.In [7], Ke and Tang established the existence of a nontrivial solution for a class of noncooperative elliptic systems with nonlinearities of sup-linear growth by using the minimax methods.In [17], Zou obtained the multiplicity of nontrivial solutions with the number of them that depends on the dimension of the eigenspaces between resonant values when the noncooperative system is resonant both at zero and at infinity by using a new abstract critical point theorem.
In this paper, we shall study the system (P) when ( ) , , F x u v is subquadratic at zero.Compared with the existing results, we do not need to make any assumptions at infinity on the nonlinearity.The nonlinearity can be subquadratic, asymptotically quadratic or superquadratic at infinity.Under this general condition, we shall prove that system (P) has infinitely many nontrivial solutions ( ) Our main tool is a variant of the Clark's theorem established by Liu and Wang [18].In [18], the authors extended the classical Clark's theorem and gave a variant of the Clark's theorem for the strongly indefinite functionals, then they used it to study the sublinear Hamiltonian systems and obtained infinitely many periodic solutions.We make the following assumptions: , , lim , Now we state our main result.Theorem 1.1.Assume that (F 1 )-(F 3 ) hold, then the system (P) has infinitely many solutions ( ) 2. Now we give a comparison between our result and some existing results.In the previous works, the authors always need to make the assumptions on the nonlinearity at infinity.For example, the sub-quadratic cases were considered in [1], the superquadratic cases were considered in [2] [4] [7] [8] [12] [14] and the asymptotically quadratic cases were considered in [6] [9] [11] [13] [15] [16] [17].Compared with these results, our nonlinearity F is general at infinity, we do not need to assume the behavior of the nonlinear term F at infinity.Besides, instead of the global symmetry condition: ) F are all sa- tisfies the conditions (F 1 )-(F 3 ), but they are subquadratic, asymptotically quadratic and superquadratic at infinity respectively.Now we give a detailed explanation of our proof.Firstly, we modify the nonlinearity.We choose a modified function F  and consider the corresponding modified elliptic system.We define the functional corresponding to this modified system.And due to the resonance of the modified system, we use a penalized functional technique and construct a penalized functional.Secondly, we show that the penalized functional satisfies the ( )

PS
* condition and is bounded from below, and prove that the penalized functional satisfies the other conditions of the Clark's theorem.Finally, by using the Clark's theorem we prove that the penalized functional has a sequence of nontrivial critical points with their L ∞ norm tending to zero.By using this fact, we conclude that they are nontrivial solutions of the system (P) for k large enough.
The paper is organized as follows.In Section 2, we give some preliminary results about the variational structure of the system (P) and a new abstract critical point theorem for the indefinite functional.In Section 3, we give the proof of our main result.

Preliminaries
In this section, we give the preliminary results and a variant of Clark's theorem to prove the Theorem 1.1.Let ( ) H Ω be the usual Sobolev space with the inner product ( ) The inner product and norm of E are given by where ( ) to denote the norm of ( ) ( ) For simplicity, we suppose that λ γ ≥ and 0 , , , , , U E ∈ .Then, the eigenvalues of the linear problem U kU =  have the form (see [2]) ( ) We only consider the case of ( ) ∉  is similar and simpler.Then let . And denote by Then L is a bounded self-adjoint linear operator on E. According to L, we can split E as are the subspaces of E on which L is positive definite, negative definite, and null.It is not difficult to see that ( ) ( ) ( ) And there exists a constant 0 c > , such that for any U E In what follows, for U E ∈ , we always write ∈Ω × , and ( ) , where . Consider the following elliptic system Define the functional J on E by By the definition of F  , it is known that ( )  and the solutions of (P') correspond to the critical points of J. Since 0 dim 0 E > , the functional J may not satisfy the compact condition.To overcome this difficulty, we use a pena-lized functional technique.Define ( ) ( ) 2 0, 0 , , , ( ) ( ) In order to prove Theorem 1.1, we introduce a variant of the Clark's theorem established by Liu and Wang [18].Let X be a Banach space, { } 0  .We say that ( ) 2) There exists 0 r > such that for any 0 a r < < there exists a critical point u such that u a = and Remark 2.2.From Theorem 2.1, it is clear that there exist a sequence of critical points 0 The additional information on the norms of the critical points is very important in the proof of Theorem 1.1.

Proof of Theorem 1.1
In this section, we shall use Theorem 2.1 to prove Theorem 1.1.Proof of theorem 1.1.Let ( ) . By (F 1 ) and the definition of J ρ , we see that = and J ρ is an even functional.Now we prove that J ρ satisfies the other assumptions of Theorem 2.1 into three steps.
Step 1.We prove that the functional J ρ satisfies the ( ) be a ( ) is bounded, and | 0 as .
Then by (3.9) and (3.10), we have that n U U → in 0 X .An the (PS) condi- tion is proved.
Step 3. We prove that there exists 0 0 n > such that for any k For any 0 T > , from (F 3 ) and the definition of F  , we can find and 0 E are finite dimensional spaces, there exists a suffi- ciently large 0 m T > , such that for every ( ) Therefore, for ( )

1 )
contains a subsequence converging to a critical point of Φ .Denote by Σ the family of closed symmetric subsets of X which do not contain 0. For A∈Σ , the genus ( )A γof A is by definition the smallest integer n for which there exists an odd and continuous mapping no such mapping exists, and ( ) 0 γ ∅ = .Theorem 2.1 ([18]).Assume that Φ is even and satisfies the ( ) least one of the following conclusions holds.There exists a sequence of critical points { } as k → ∞ .

1 ,
J ρ has a sequence of nontrivial critical points { } we only need the local symmetry condition (F 2 ).