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On a class of singular elliptic problems with the perturbed Hardy-Sobolev operator. (English) Zbl 1258.35104

Summary: In this paper, firstly, we investigate a class of singular eigenvalue problems with the perturbed Hardy-Sobolev operator, and obtain some properties of the eigenvalues and the eigenfunctions. (i.e. existence, simplicity, isolation and comparison results). Secondly, applying these properties of eigenvalue problem, and the linking theorem for two symmetric cones in Banach space, we discuss the following singular elliptic problem \[ \left\{\begin{aligned} & -\Delta_{p}u-a(x)\frac{|u|^{p-2}u}{|x|^{p}}= \lambda \eta(x)|u|^{p-2}u+ f(x,u) \quad x \in \Omega,\\ & u =0 \quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad x\in\partial \Omega, \end{aligned}\right. \] where \({a(x)=(\frac{n-p}{p})^{p}q(x),}\) if \(1 < p < n, {a(x)=(\frac{n-1}{n})^{n} \frac{q(x)}{({\mathrm log}\frac{R}{|x|})^{n}},}\) if \(p = n\), and prove the existence of a nontrivial weak solution for any \({\lambda \in \mathbb{R}.}\)

MSC:

35J75 Singular elliptic equations
35P05 General topics in linear spectral theory for PDEs
35D30 Weak solutions to PDEs
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