Quasilinear nonlocal elliptic problems with variable singular exponent

Abstract

In this article, we provide existence results to the following nonlocal equation \begin{document}$ \begin{align*} \begin{cases} (-\Delta)_p^{s} u = g(x,u),\;u>0\; \text{in}\; u = 0 \; \text{in}\; \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*}\quad\quad(P_ \lambda)$ \end{document} where \begin{document}$ (-\Delta)_{p}^{s} $\end{document} is the fractional \begin{document}$ p $\end{document} -Laplacian operator. Here \begin{document}$ \Omega \subset \mathbb R^N $\end{document} is a smooth bounded domain, \begin{document}$ s\in(0,1) $\end{document} , \begin{document}$ p>1 $\end{document} and \begin{document}$ N>sp $\end{document} . We establish existence of at least one weak solution for \begin{document}$ (P_ \lambda) $\end{document} when \begin{document}$ g(x,u) = f(x)u^{-q(x)} $\end{document} and existence of at least two weak solutions when \begin{document}$ g(x,u) = \lambda u^{-q(x)}+ u^{r} $\end{document} for a suitable range of \begin{document}$ \lambda>0 $\end{document} . Here \begin{document}$ r\in(p-1,p_{s}^{*}-1) $\end{document} where \begin{document}$ p_s^{*} $\end{document} is the critical Sobolev exponent and \begin{document}$ 0 .