Existence and Multiplicity of Solutions for a Nonlocal Problem with Critical Sobolev–Hardy Nonlinearities

Abstract

The purpose of this paper is to study the nonlocal elliptic equation involving critical Hardy–Sobolev exponents as follows,

$$\begin{aligned}(\mathrm{P}) {\left\{ \begin{array}{ll} (-\Delta )^s u -\mu \frac{u}{|x|^{2s}}= \lambda |u|^{q-2}u +\frac{|u|^{2_\alpha ^*-2}u}{|x|^\alpha } &{} \text {in} \ \Omega ,\\ u=0 &{} \text {in} \ \mathbb {R}^n\setminus \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega \subset \mathbb {R}^N$ is a bounded domain with Lipschitz boundary, $0<s<1$, $\lambda >0$ is a parameter, $0\le \mu <\mu _0,$ with $\mu _0=4^s\frac{\Gamma ^2\left( \frac{N+2s}{4}\right) }{\Gamma ^2\left( \frac{N-2s}{4}\right) }$ being the sharp constant of the fractional Hardy–Sobolev in ${\mathbb R}^N,$ $0< \alpha<2s<N$, $1<q <2_s^*$ where $2_s^* = \frac{2N}{N-2s}$ and $2_\alpha ^* = \frac{2(N-\alpha )}{N-2s}$ are the fractional critical Sobolev and Hardy–Sobolev exponents respectively. The fractional Laplacian $(-\Delta )^s$ with $s \in (0,1)$ is the non linear non local operator defined on smooth functions by:

$$\begin{aligned} (-\Delta )^s u(x)=-\frac{1}{2} \int _{\mathbb {R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}\,\mathrm{d}y,\quad \text {for all } x \in \mathbb {R}^N. \end{aligned}$$

We combine sub and super-solution method combine with min-max method in order to prove the existence and multiplicity of solutions to the problem $(\mathrm{P}).$