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On the eigenvalue problem for the Hardy-Sobolev operator with indefinite weights. (English) Zbl 1125.35332
Summary: 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 \(1<p\leq N\), \(\Omega\) is a bounded domain containing \(0\) in \(\mathbb R^N\), \(\Delta_{p}\) is the \(p\)-Laplacian, 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, isolation of the first eigenvalue, nodal domain properties. Furthermore we show the existence of a nontrivial curve in the Fučik spectrum.

MSC:
35J20 Variational methods for second-order elliptic equations
35J70 Degenerate elliptic equations
35P05 General topics in linear spectral theory for PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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