Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 33, pp. 1-9.
Title: Eigenvalue problems for the p-Laplacian with indefinite weights
Author: Mabel Cuesta (Univ. du Littoral, Calais, France)
Abstract:
We consider the eigenvalue problem
$-\Delta_p u=\lambda V(x) |u|^{p-2} u, u\in W_0^{1,p} (\Omega)$
where $p>1$, $\Delta_p$ is the p-Laplacian operator,
$\lambda >0$, $\Omega$ is a bounded domain in $\mathbb{R}^N$ and $V$ is a
given function in $L^s (\Omega)$ ($s$ depending on $p$ and $N$).
The weight function $V$ may change sign and has nontrivial positive part.
We prove that the least positive eigenvalue is simple, isolated in the
spectrum and it is the unique eigenvalue associated to a nonnegative
eigenfunction. Furthermore, we prove the strict monotonicity of the
least positive eigenvalue with respect to the domain and the weight.
Submitted April 4, 2001. Published May 10, 2001.
Math Subject Classifications: 35J20, 35J70, 35P05, 35P30
Key Words: Nonlinear eigenvalue problem; p-Laplacian; indefinite weight.