Electron. J. Diff. Equ., Vol. 2011 (2011), No. 64, pp. 1-22.

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

T. V. Anoop

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.

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T. V. Anoop
The Institute of Mathematical Sciences
Chennai 600113, India
email: tvanoop@imsc.res.in

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