Electron. J. Differential Equations, Vol. 2017 (2017), No. 149, pp. 1-27.

Asymmetric superlinear problems under strong resonance conditions

Leandro Recova, Adolfo Rumbos

We study the existence and multiplicity of solutions of the problem
 -\Delta u = -\lambda_1 u^- + g(x,u),\quad \text{in } \Omega; \cr
 u = 0, \quad \text{on } \partial\Omega,
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ $(N\geq 2)$, $u^-$ denotes the negative part of $u\colon\Omega\to\mathbb{R}$, $\lambda_1$ is the first eigenvalue of the N-dimensional Laplacian with Dirichlet boundary conditions in $\Omega$, and $g:\Omega\times\mathbb{R}\to\mathbb{R}$ is a continuous function with $g(x,0) =0$ for all $x\in\Omega$. 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 [14].

Submitted February 19, 2017. Published June 23, 2017.
Math Subject Classifications: 35J20.
Key Words: Strong resonance; Palais-Smale condition; Ekeland's principle.

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Leandro L. Recova
Ericsson Inc.,
Irvine, California 92609, USA
email: leandro.recova@ericsson.com
Adolfo J. Rumbos
Department of Mathematics
Pomona College
Claremont, California 91711, USA
email: arumbos@pomona.edu

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