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.