On the de Th\'elin eigenvalue problem and Landesman-Lazer conditions for quasilinear systems

TL;DR

This paper establishes the simplicity and isolation of the smallest eigenvalue for a quasilinear elliptic system, characterizes a sequence of eigenvalues variationally, and proves existence of solutions under new Landesman-Lazer conditions.

Contribution

It extends previous results by characterizing eigenvalues variationally and establishing existence of solutions in resonance with new Landesman-Lazer conditions.

Findings

Smallest eigenvalue is simple and isolated.

Eigenvalues are characterized variationally.

Existence of solutions under new Landesman-Lazer conditions.

Abstract

In this paper we prove that the smallest eigenvalue $\lambda_1$ of the eigenvalue problem for a quasilinear elliptic systems introduced by de Th\'elin in \cite{DT}, is not only simple (in a suitable sense), but also isolated. Moreover, we characterize variationally a sequence $\{\lambda_k\}_k$ of eigenvalues, taking into account a suitable deformation lemma for $C^1$ submanifolds proved in \cite{BON}. Furthermore we prove the existence of a weak solution for a quasilinear elliptic systems in resonance around $\lambda_1$, under new sufficient Landesman-Lazer type conditions, extending the results by Arcoya and Orsina \cite{AO}.