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

In this paper we study the eigenvalue problem -Delta(p)u - a(x)vertical bar u vertical bar(p-2)u = lambda vertical bar u vertical bar(p-2)u, u is an element of W-0(1,p)(Omega), where 1 < p <= N, Omega is a bounded domain containing 0 in R-N, Delta(p) is the p-Laplacean, and a(x) is a function related to Hardy-Sobolev inequality. The weight function V (x) is an element of 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.