A fractional p-Laplacian problem with multiple critical Hardy-Sobolev nonlinearities

Abstract

In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities, \begin{align*} (-Δ_p)^su - μ\dfrac{\vert u \vert^{p-2} u}{\vert x \vert^{ps}} = \dfrac{\vert u\vert^{p^*_s(β)-2}u}{\vert x \vert^β} + \dfrac{\vert u\vert^{p^*_s(α)-2} u}{\vert x \vert^α}, \end{align*} where $x\in {\mathbb{R}}^N$, $u\in D^{s,p}({\mathbb{R}}^N)$, $0<s<1$, $1<p<+\infty$, $N>sp$, $0<\alpha <sp$, $0<\beta <sp$, $\beta \ne \alpha$, $\mu <{\mu }_H:={\inf }_{u\in D^{s,p}({\mathbb{R}}^N)\backslash \{0\}}[u{]}_{s,p}^p/||u|{|}_{s,p}^p>0$. To prove the existence of solution to the problem we have to formulate a refined version of the concentration-compactness principle and, as an independent result, we have to show that the extremals for the Sobolev inequality are attained.