Electron. J. Diff. Equ., Vol. 2011 (2011), No. 145, pp. 1-11.

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

Evandro Monteiro

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 2, 2011.
Math Subject Classifications: 35J30, 35J35.
Key Words: Asymptotically linear; Morse theory; shifting theorem; multiplicity of solutions.

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Evandro Monteiro
UNIFAL-MG, Rua Gabriel Monteiro da Silva, 700. Centro
CEP 37130-000 Alfenas-MG, Brazil
email: evandromonteiro@unifal-mg.edu.br

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