On the de Thélin eigenvalue problem and Landesman-Lazer conditions for quasilinear systems

Abstract

In this paper we prove that the smallest eigenvalue ${\lambda }_1$ of the eigenvalue problem for a quasilinear elliptic systems introduced by de Thélin 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}.

To appear on Journal of Differential Equations