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.