Electron. J. Diff. Eqns., Vol. 2001(2001), No. 33, pp. 1-9.

Eigenvalue problems for the p-Laplacian with indefinite weights

Mabel Cuesta

We consider the eigenvalue problem
$-\Delta_p u=\lambda V(x) |u|^{p-2} u, u\in W_0^{1,p} (\Omega)$
where $p greater than 1$, $\Delta_p$ is the p-Laplacian operator, $\lambda greater than 0$, $\Omega$ 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.

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Mabel Cuesta
Universite du Littoral
ULCO, 50, rue F. Buisson,
B.P. 699, F-62228 Calais, France
e-mail: cuesta@lmpa.univ-littoral.fr

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