Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 64, pp. 1-22.

Title: Weighted eigenvalue problems for the p-Laplacian with
       weights in weak Lebesgue spaces
       
Author: T. V. Anoop (Institute of Math. Sciences, Chennai, India)

Abstract:
 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.

Submitted November 11, 2010. Published May 17, 2011.
Math Subject Classifications: 35J92, 35P30, 35A15.
Key Words: Lorentz spaces; principal eigenvalue; radial symmetry;
           Ljusternik-Schnirelmann theory.