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The solvability of fractional elliptic equation with the Hardy potential. (English) Zbl 1435.35408

Summary: In this paper, we study the existence and nonexistence of solutions to fractional elliptic equations with the Hardy potential \(\begin{cases} ( - \Delta)^s u - \lambda \left( u / \left| x\right|^{2 s}\right) = u^{r - 1} + \delta g \left( u\right) , & \text{in } \Omega, \\ u \left( x\right) > 0 , & \text{in } \Omega, \\ u \left( x\right) = 0 , & \text{in } \mathbb{R}^N \smallsetminus \Omega, \end{cases}\) where \(\Omega\subset \mathbb{R}^N\) is a bounded Lipschitz domain with \(0\in\Omega\), \((-\Delta)^s\) is a fractional Laplace operator, \(s\in(0,1)\), \(N>2s\), \(\delta\) is a positive number, \(2<r<r(\lambda, s)\equiv( N + 2 s - 2 \alpha_\lambda / N - 2 s - 2 \alpha_\lambda)+1\), \(\alpha_\lambda\in(0,(( N - 2 s) / 2))\) is a parameter depending on \(\lambda\), \(0<\lambda<\Lambda_{N , s} \), and \(\Lambda_{N , s}= 2^{2 s}( \Gamma^2 (( N + 2 s) / 4))/(\Gamma^2 ((N - 2s) / 4))\) is the sharp constant of the Hardy-Sobolev inequality.

MSC:

35R11 Fractional partial differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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