Weighted eigenvalue problems for the \(p\)-Laplacian with weights in weak Lebesgue spaces

Anoop, T.V.

Electronic journal of differential equations, Tome 2011 (2011)

Résumé

We consider the nonlinear eigenvalue problem

$ -\Delta_p u= \lambda g |u|^{p-2}u,\quad u\in \mathcal{D}^{1,p}_0(\Omega) $

where $\Delta_p$ is the p-Laplacian operator, $\Omega$ is a connected domain in $\mathbb{R}^N$ with $N>p$ and the weight function g is locally integrable. We obtain the existence of a unique positive principal eigenvalue for g such that $g^+$ lies in certain subspace of weak- $L^{N/p}(\Omega)$. The radial symmetry of the first eigenfunctions are obtained for radial g, when $\Omega$ is a ball centered at the origin or $\mathbb{R}^N$. The existence of an infinite set of eigenvalues is proved using the Ljusternik-Schnirelmann theory on $\mathcal{C}^1$ manifolds.

Détail

Classification : 35J92, 35P30, 35A15
Keywords: Lorentz spaces, principal eigenvalue, radial symmetry, Ljusternik-Schnirelmann theory