Leandro Recova, Adolfo Rumbos
We study the existence and multiplicity of solutions of the problem
where is a smooth bounded domain in , denotes the negative part of , is the first eigenvalue of the N-dimensional Laplacian with Dirichlet boundary conditions in , and is a continuous function with for all . We assume that the nonlinearity g(x,s) has a strong resonant behavior for large negative values of s and is superlinear, but subcritical, for large positive values of s. Because of the lack of compactness in this kind of problem, we establish conditions under which the associated energy functional satisfies the Palais-Smale condition. We prove the existence of three nontrivial solutions of the problem as a consequence of Ekeland's Variational Principle and a variant of the mountain pass theorem due to Pucci and Serrin .
Submitted February 19, 2017. Published June 23, 2017.
Math Subject Classifications: 35J20.
Key Words: Strong resonance; Palais-Smale condition; Ekeland's principle.
Show me the PDF file (377 KB), TEX file for this article.
| Leandro L. Recova|
Irvine, California 92609, USA
| Adolfo J. Rumbos |
Department of Mathematics
Claremont, California 91711, USA
Return to the EJDE web page