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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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