Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 145, pp. 1-11.

Title: Asymptotically linear fourth-order elliptic problems 
       whose nonlinearity crosses several eigenvalues

Author: Evandro Monteiro (UNIFAL-MG, Alfenas-MG, Brazil)

Abstract: 
 In this article we prove the existence of multiple solutions for
 the fourth-order elliptic problem
 $$\displaylines{
 \Delta^2u+c\Delta u = g(x,u) \quad\hbox{in }  \Omega\cr
 u =\Delta u= 0  \quad\hbox{on }  \partial \Omega,
 }$$
 where $\Omega \subset \mathbb{R}^N$ is a bounded domain,
 $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a function
 of class $C^1$ such that $g(x,0)=0$ and it
 is asymptotically linear at infinity.
 We study the cases when  the parameter c is less than the
 first eigenvalue,  and between two consecutive eigenvalues
 of the Laplacian. To obtain solutions we use the Saddle Point Theorem,
 the Linking Theorem, and Critical Groups Theory.

Submitted February 15, 2011. Published November 02, 2011.
Math Subject Classifications: 35J30, 35J35.
Key Words: Asymptotically linear; Morse theory; shifting theorem;
           multiplicity of solutions.