A fractional p-Laplacian problem with multiple critical Hardy-Sobolev nonlinearities
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$, $u\in D^{s,p}(\mathbb{R}^N)$, $0 sp$, $0<α 0$. 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.
math.AP↗