On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Let $Omega$ be a bounded open set of $mathbb R^n$, $nge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $lambda_2(p,Omega)$ of the anisotropic $p$-Laplacian [ -mathcal Q_pu:=- extrmdiv left(F^p-1( abla u)F_\xi ( abla u) ight), ] where $F$ is a suitable smooth norm of $mathbb R^n$ and $pin]1,+infty[$. We provide a lower bound of $lambda_2(p,Omega)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p o+infty$.