Existence of a Nontrivial Solution for a Class of Superquadratic Elliptic Problems *

We consider the existence of a nontrivial solution for the Dirichlet boundary value problem     = , , in , = 0, . x u u  on u a x u g       We prove an abstract result on the existence of a critical point for the functional f on a Hilbert space via the local linking theorem. Different from the works in the literature, the new theorem is constructed under the condition instead of condition.   C   PS   C


Introduction and Main Results
Consider the Dirichlet boundary value problem where and is a bounded domain whose boundary is a smooth manifold.

 
We assume that , where 0 .In [1], Li and Willem established the existence of a nontrivial solution for problem (1) under the following well-known Ambrosetti-Rabinowitz superlinearity condition: there exists for all u L  x   and , which has been used extensively in the literature; see [1][2][3][4] and the references therein.It is easy to see that condition (AR) does not include some superquadratic nonlinearity like   In [5], Qin Jiang and Chunlei Tang completed the Theorem 4 in [1], and obtained the existence of a nontrivial solution for problem (1) under a new superquadratic condition which covered the case of (G0).The conditions are as follows: , as uniformly on , and , If 0 is an eigenvalue of (with Dirichlet boundary condition) assume also the condition that:

1)
, for all , for all Note that (G4) is also (AR) type condition.The aim of this paper is to consider the nontrivial solution of problem (1) in a more general sense.Without the Ambrosetti-Rabinowitz superlinearity condition (AR) or (G4), the superlinear problems become more complicated.We do not know in our situations whether the (PS) or sequence are bounded.However, we can check that any Cerami (or ) sequence is bounded.
The definition of  (or ) sequence can be found in [6].

PS
We will obtain the same conclusion under the condition instead of condition.So we only need the following conditions instead of (G3) (G4): satisfies conditions of (G1) (G2) (G5) and (G3').

Dirichlet boundary condition). Then problem (1) has at least one nontrivial solution.
Remark 1.There are many functions which are superlinear but it is not necessary to satisfy Ambrosetti-Rabinowitz condition.For example, where  . Then it is easy to check that (AR) does not hold.On the other hand, in order to verify (AR), it usually is an annoying task to compute the primitive function of and sometimes it is almost impossible.For example, where  .Remark 2. Our condition is much weaker than (AR) type condition (cf.[6]).

Proof of Theorem
Define a functional f in the space by is the space spanned by the eigenvectors corresponding to negative (positive) eigenvalue of   . In this paper, we shall use the following local linking theorem (Lemma 2.1) to prove our Theorem .Let X be a real Banach space with  satisfies the following assumptions: Then f has at least two critical points.Proof of Theorem 1.We shall apply Lemma 2.1 to the functional f associated with (1), we consider the case where 0 is an eigenvalue of and The other case are similar.
f maps bounded sets into bounded sets.
Hence f C X R  and maps bounded sets into bounded sets.
In fact, 2) f has a local linking at 0. It follows from (G2) and (G3) that, for any > 0  , there exists , such that > 0 , , we obtain, on 2 X , for some Decompose 1 X into V when , .Also set .Since V is a finitedimensional space, there exists , such that

On
, we have, by ( 5)  On , we have also by ( 5)   and for some For n large, from assumption (G3'), with is bounded.
, , Arguing indirectly, assume n .Set 1, s  .In addition, using (6)     = 0 f u  4) Finally, we claim that, for every , m N (I)  we see that   u n is bounded in X , going if necessary to a subsequence, we can assume that n