Multiplicity results of fractional-Laplace system with sign-changing and singular nonlinearity

Abstract

In this article, we study the following fractional-Laplacian system with singular nonlinearity \begin{equation*} (P_{\lambda,\mu}) \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda f(x) u^{-q}+ \frac{\alpha}{\alpha+\beta}b(x) u^{\alpha-1} w^\beta\; \text{in}\;\Omega \\ (-\Delta)^s w = \mu g(x) w^{-q}+ \frac{\beta}{\alpha+\beta} b(x) u^{\alpha} w^{\beta-1}\; \text{in}\;\Omega \\ \quad \quad u, w>0\;\text{in}\;\Omega, \quad u = w = 0 \; \mbox{in}\; \mathbb{R}^n \setminus\Omega, \end{array} \quad \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n>2s$, $s\in(0,1)$, $0<q<1$, $\alpha>1$, $\beta>1$ satisfy $2<\alpha+\beta< 2_{s}^*-1$ with $2_{s}^*=\frac{2n}{n-2s}$, the pair of parameters $(\lambda,\mu) \in \mathbb{R}^2\setminus\{(0,0)\}$. The weight functions $f,g: \Omega\subset\mathbb{R}^n \to \mathbb{R}$ such that $0<f$, $g\in L^{\frac{\alpha+\beta}{\alpha+\beta-1+q}}(\Omega)$, and $b:\Omega\subset \mathbb{R}^n \to \mathbb{R}$ is a sign-changing function such that $b(x)\in L^{\infty}(\Omega)$. Using variational methods, we show existence and multiplicity of positive solutions of $(P_{\lambda,\mu})$ with respect to the pair of parameters $(\lambda,\mu)$.