Special Issue in honor of Alan C. Lazer. Electron. J. Diff. Eqns., Special Issue 01 (2021), pp. 183-202.

An asymmetric problem at resonance with a one-sided Ahmad-Lazer-Paul condition

Leandro L. Recova, Adolfo J. Rumbos

Abstract:
In this article, we study the semilinear elliptic boundary value problem

where $u^{-}$ denotes the negative part of $u:\Omega\to \mathbb{R}$; $\lambda_1$ is the first eigenvalue of the N-dimensional Laplacian with Dirichlet boundary conditions in a connected, open, bounded set $\Omega\subset\mathbb{R}^N$, $N\geq 2$; and $g{:} \overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is a continuous function. Assuming a one-sided Ahmad-Lazer-Paul condition, we establish conditions for existence and multiplicity of solutions by using variational methods and infinite-dimensional Morse Theory.

Published October 6, 2021.
Math Subject Classifications: 35J20.
Key Words: Resonance; critical groups; Morse theory; Ekeland's variational principle.
DOI: https://doi.org/10.58997/ejde.sp.01.r2

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Leandro L. Recôva
T-Mobile Inc.,
Ontario, California 91761, USA
email leandro.recova3@t-mobile.com
Adolfo J. Rumbos
Department of Mathematics
Pomona College
Claremont, California 91711, USA
email: arumbos@pomona.edu

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