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

Abstract

We consider the nonlinear eigenvalue problem $$ -Delta_p u= lambda g |u|^{p-2}u,quad uin mathcal{D}^{1,p}_0(Omega) $$ 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.

Keywords

• Lorentz spaces
• principal eigenvalue
• radial symmetry
• Ljusternik-Schnirelmann theory