Multiple Solutions for a Class of Semilinear Elliptic Equations with Nonlinear Boundary Conditions

In this paper, using Local Linking Theorem, we obtain the existence of multiple solutions for a class of semilinear elliptic equations with nonlinear boundary conditions, in which the nonlinearites are compared with higher Neumann eigenvalue and the first Steklov eigenvalue.


Introduction
In this paper, we investigate the multiple solutions for semilinear elliptic equation with nonlinear boundary conditions , , is bounded domain with smooth boundary  and       is the outward normal derivative on  , and the function   , and 0 a.e. on with d 0.
 Problems of the above type have been discussed extensively.In 1902, Steklov (see [1]) studied the eigenvalue problem 0, , Auchmuty (see [2]) considered the eigenvalue problem where   c x satisfies the condition C) and proved that the eigenfunctions provide a complete orthonormal bases of certain closed subspace of   1 H  .Using sub and super-solutions method, Amann (see [3]), Mawhin and Schmitt (see [4]) obtained some existence results for the problem (1.1).However, since it is based on comparison techniques, the sub and super-method does not apply when the nonlinearities are compared with higher eigenvalues.
In this paper, using the Local Linking Theorem, we obtain multiple solutions for the problem (1.1), which the nonlinearites are compared with higher Neumann eigenvalue and the first Steklov eigenvalue.

Let
with the associated norm and the associated norm  c x satisfies the condition C), by Equation (2.2), we can split as a direct orthogonal sum.Now, we state the Local Linking theorem introduced by [5].
1) there exists a constant 0 2)  is bounded below and inf 0, X   then the functional  has at least two nontrivial critical points.Proof.See Theorem 4 in [5].For the problem (1.3), Auchmuty (see [2]) obtained that  is the first Steklov eigenvalue for the problem (1.3).In [6], for the Neu- they obtain that the above problem has a sequence of real eigenvalues 1 2 0 , as , with finite dimensional eigenspaces.Assume that, : for all t   and for a.e. ; x   H2) There exist 2 0 c  and for all t   and for a.e. ; x   H3) There exist , 0 H4) There exist a integer 1, 0 k    and four constants , , , a b ), and H1)-H4) hold, Then the problem (1.1) has at least two distinct nontrivial solutions.

The Proofs of Theorem 2.2
Now, we define the functional N N L   , the continuity of the trace operator from Furthermore, the functional   J u is weakly continuous, and 2) and a simple computation, we obtain that the critical point of the functional is the weak solution of the problem (1.1).Lemma 3.1 (see [7]) Assume that the function   c x satisfies the condition C), H1) and H2) hold.If   H  Proof By H3), we obtain that there exist some constants 0 , From H1), H2) and Equation (3.3), we obtain that there exist Hence, we obtain that Assume that , c u   then using the continuity of the trace operator from by Equation (3.5), we obtain By H3), we obtain 1 1 Hence we obtain that the functional   u  is bounded from below, and every satisfies (PS) condition and is bounded from below.The Proof of Theorem 2.2 We write , where Hence, we have is a finite dimensional space, by [2], we obtain that for given 0, , by H4), we have We have for 0 Therefore, we obtain that On the other hand, let Combining Equation (3.4), Equation (3.6), H1) and H2), we have From Equation (2.6), we have, for 0 Since 1 2 , 2 q q  , we can choose 2 0   sufficiently small and

Conclusion
Using Local Linking Theorem, we obtain the existence of two nontrival weak solutions for the problem (1.1)

1 u.
As the function   c x satisfies the condition C), we define the weighted

cu. 1 u
By Corollary 3.3 in[2], we obtain that the norm c u is equivalent to the standard norm .As the function( ) and the Holder inequality, we obtain that the functional   u  is well defined.Moreover, by Lemma 2.1, and Lemma 4.2 in [7], we obtain that

Lemma 3 . 2
PS) sequence for the functional  , and   Assume that   c x satisfies the condition C), and H1)-H3) hold, the functional

1 ,
 possesses two nonzero critical point.From (3.3), we obtain that there exist two nontrivial weak solutions for the problem (1.1).