Two disjoint and infinite sets of solutions for a nonlocal problem involving a Hardy potential and critical growth with concave nonlinearities

Abstract

In this paper, we consider the following problem involving fractional Laplacian with a Hardy potential operator                                          

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{\alpha } u -\mu \dfrac{u}{\vert x\vert ^{2\alpha }}=\vert u\vert ^{2_{\alpha }^{*}-2} u+\lambda \vert u\vert ^{q-2}u, &{} \text{ in } \Omega , \\ u=0, &{} \text{ on } \partial \Omega , \end{array}\right. \end{aligned}$$

where $\Omega$ is a smooth bounded domain in ${\mathbb {R}}^{N}$ which contains the origin, $\mu \ge 0$, $\lambda >0$, $0<\alpha <1$, $1< q < 2$, $2_{\alpha }^{*}=\frac{2 N}{N-2 \alpha }$ and $(-\Delta )^{\alpha }$ is the spectral fractional Laplacian. Using an approximate argument, some local Pohozaev Identities, Fountain Theorem and its Dual version, we prove that the above problem has two disjoint and infinite sets of solutions under suitable conditions on N, q and $\mu$.