Fractional Hardy-Sobolev elliptic problems

Abstract

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-Δ)^{\frac α2} u=λ|u|^{r-2}u+μ\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad Ω, \\ \\ u=0 &{\rm on }\quad \partialΩ, \end{array} \right. \end{equation} where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$ with $0\in \Omega$, $\lambda ,\mu >0,0<s\le \alpha$, $(-\Delta {)}^{\frac{\alpha }{2}}$ is the fractional Laplacian operator with $0<\alpha <2$. We establish existence results of problem \eqref{eq:1} for subcritical, Sobolev critical and Hardy-Sobolev critical cases.

21 pages

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