Multiplicity results of fractional $p$-Laplace equations with sign-changing and singular nonlinearity

Abstract

In this article, we study the following fractional $p$-Laplacian equation with singular nonlinearity \begin{equation*} (P_{\la}) \left\{ \begin{array}{lr} - 2\int_{\mb R^n}\frac{|w(y)-w(x)|^{p-2}(w(y)-w(x))}{|x-y|^{n+ps}}dy = a(x) w^{-q}+ \la b(x) w^r\; \text{in}\; \Om \quad \quad w>0\;\text{in}\;\Om, \quad w = 0 \; \mbox{in}\; \mb R^n \setminus\Om, \end{array} \quad \right. \end{equation*} where $\Om$ is a bounded domain in $\mb R^n$ with smooth boundary $\partial \Om$, $n> ps$,$s\in(0,1)$, $\la>0$, $0<q<1$, $q<p-1<r< p_{s}^*-1$ with $p_{s}^*=\frac{np}{n-ps}$, $a: \Om\subset\mb R^n \ra \mb R$ such that $0< a(x)\in L^{\frac{p^{*}_{s}}{p^{*}_{s}-1+q}}(\Om)$, and $b:\Om\subset\mb R^n \ra \mb R$ is a sign-changing function such that $b(x)\in L^{\frac{p^{*}_{s}}{p^{*}_{s}-1-r}}(\Om)$. Using variational methods, we show existence and multiplicity of positive solutions of $(P_{\la})$ with respect to the parameter $\la$.