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 $$\displaylines{ (-\Delta)^s u = \lambda f(x) u^{-q} + \frac{\alpha}{\alpha+\beta}b(x) u^{\alpha-1} w^\beta\quad \text{in }\Omega \cr (-\Delta)^s w = \mu g(x) w^{-q}+ \frac{\beta}{\alpha+\beta} b(x) u^{\alpha} w^{\beta-1}\; \text{in } \Omega \cr u, w>0\text{ in }\Omega, \quad u = w = 0 \text{ in } \mathbb{R}^n \setminus\Omega, }$$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n>2s$, $s\in(0,1)$, $01$ satisfy $2