Perturbed asymptotically linear problems

TL;DR

This paper studies the existence and multiplicity of solutions for perturbed semilinear elliptic problems with asymptotically linear nonlinearities, using variational and topological methods to analyze stability under perturbations.

Contribution

It introduces new techniques to prove solution existence and multiplicity for perturbed problems lacking variational structure, extending previous results to continuous perturbations.

Findings

Existence of solutions under subcritical, asymptotically linear conditions.

Stability of critical levels under small perturbations.

Multiplicity results for odd nonlinearities in resonant and non-resonant cases.

Abstract

The aim of this paper is investigating the existence of solutions of some semilinear elliptic problems on open bounded domains when the nonlinearity is subcritical and asymptotically linear at infinity and there is a perturbation term which is just continuous. Also in the case when the problem has not a variational structure, suitable procedures and estimates allow us to prove that the number of distinct crtitical levels of the functional associated to the unperturbed problem is "stable" under small perturbations, obtaining multiplicity results if the nonlinearity is odd, both in the non--resonant and in the resonant case.