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

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}.