Fractional Hardy-Sobolev elliptic problems. (English) Zbl 1437.35376

Yang, Jianfu; Yu, Jian

Topol. Methods Nonlinear Anal. 55, No. 1, 257-280 (2020).

Summary: In this paper, we study the following singular nonlinear elliptic problem \[\begin{cases} (-\Delta)^{ \alpha/ 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^s} &\text{in }\Omega, \\ u=0 &\text{on } \partial\Omega, \end{cases} \tag{P}\] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) (\(N\geq 2\)) with \(0\in \Omega\), \(\lambda,\mu> 0\), \(0< s\leq\alpha\), \((-\Delta)^{\alpha/ 2}\) is the spectral fractional Laplacian operator with \(0< \alpha< 2\). We establish existence results and nonexistence results of problem (P) for subcritical, Sobolev critical and Hardy-Sobolev critical cases.

MSC:

35J75	Singular elliptic equations
35R11	Fractional partial differential equations
35B33	Critical exponents in context of PDEs
35A01	Existence problems for PDEs: global existence, local existence, non-existence

Keywords:

fractional Laplacian; critical Hardy-Sobolev exponent; existence of solutions

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