Existence results for singular elliptic problem involving a fractional p-Laplacian

Abstract

In this article, the problems to be studied are the following \leqnomode \begin{equation*} \label{p} \left\{\begin{array}{ll} (-\Delta )_p^s u \pm \dfrac{|u|^{p-2}u}{|x|^{sp}} = \lambda f(x,u) & \quad \mbox{in }\ \Omega\\[0.3cm] u= 0 & \quad \mbox{on }\ \mathbb{R}^N \setminus \Omega,\tag{P$_{\pm}$} \end{array} \right. \end{equation*} \reqnomode where $\Omega$ is a bounded regular domain in $\mathbb{R}^N(N\geq 2)$ containing the origin, $p>1$, $s\in(0,1)$, $(N>ps)$, $\lambda>0$, $f : \Omega \times \mathbb{R} \longrightarrow \mathbb{R}$ is a Carath\'eodory function satisfying a suitable growth condition and $(-\Delta )_p^s$ is the fractional p-Laplacian defined as $$(-\Delta )_{p}^{s} u(x) = \displaystyle 2 \lim_{\varepsilon \rightarrow 0} \int_{\mathbb{R}^N \setminus B_{\varepsilon}(x)} \dfrac{\vert u(x)-u(y) \vert^{p-2}(u(x)-u(y))}{\vert x-y \vert^{N+sp}} ~dy, ~~~~ x \in \mathbb{R}^N,$$ where $B_{\varepsilon}(x)$ is the open $\varepsilon$-ball of centre $x$ and radius $\varepsilon$. Using the critical point theory combining to the fractional Hardy inequality, we show that the problem $(P_+)$ admits at least two distinct nontrivial weak solutions. For the problem $(P_-),$ we use the concentration-compactness principle for fractional Sobolev spaces to give a weak lower semicontinuity result and prove that problem $(P_-)$ admits at least one non-trivial weak solution.