A singular elliptic problem involving fractional $p$-Laplacian and a discontinuous critical nonlinearity

Abstract

In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob} (-\Delta)_p^su&=\mu g(x,u)+\frac{\lambda}{u^\gamma}+H(u-\alpha)u^{p_s^*-1},~\text{in}~\Omega u&>0,~\text{in}~\Omega, u&=0,~\text{in}~\mathbb{R}^N\setminus\Omega, \end{split} \end{align} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $s\in (0,1)$, $2<p<\frac{N}{s}$, $\gamma\in (0,1)$, $\lambda,\mu>0$, $\alpha\geq 0$ is real, $H$ is the Heaviside function, i.e. $H(a)=0$ if $a\leq 0$, $H(a)=1$ if $a>0$ and $p_s^*=\frac{Np}{N-sp}$ is the fractional critical Sobolev exponent. Under suitable assumptions on the function $g$, we prove the existence of solution to the problem. Furthermore, we show that as $\alpha\rightarrow0^+$, the sequence of solutions of $\eqref{main_prob}$ for each such $\alpha$ converges to a solution of the problem for which $\alpha=0$.