Electron. J. Diff. Eqns., Vol. 2002(2002), No. 33, pp. 1-12.

On the eigenvalue problem for the Hardy-Sobolev operator with indefinite weights

K. Sreenadh

Abstract:
In this paper we study the eigenvalue problem
$$ -\Delta_{p}u-a(x)|u|^{p-2}u=\lambda |u|^{p-2}u, \quad u\in W^{1,p}_{0}(\Omega), $$
where 1 less than $p\le N$, $\Omega$ is a bounded domain containing 0 in $\mathbb{R}^N$, $\Delta_{p}$ is the p-Laplacean, and $a(x)$ is a function related to Hardy-Sobolev inequality. The weight function $V(x)\in L^{s}(\Omega)$ may change sign and has nontrivial positive part. We study the simplicity, isolatedness of the first eigenvalue, nodal domain properties. Furthermore we show the existence of a nontrivial curve in the Fucik spectrum.

Submitted October 23, 2001. Published April 2, 2002.
Math Subject Classifications: 35J20, 35J70, 35P05, 35P30.
Key Words: p-Laplcean, Hardy-Sobolev operator, Fucik spectrum, Indefinite weight.

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Konijeti Sreenadh
Department of Mathematics
Indian Institute of Technology
Kanpur 208016, India.
e-mail: snadh@iitk.ac.in

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