Critical growth fractional elliptic problem with singular nonlinearities

Abstract: In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity $$\quad (-\Delta)^s u = \lambda a(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; \Omega,\quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminus\Omega,$$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n > 2s,\; s \in (0,1),\; \lambda >0,\; 0 < q \leq 1 $, $\theta \leq a(x) \in L^\infty(\Omega)$, for some $\theta>0$ and $2^*_s=\frac{2n}{n-2s}$. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $\lambda$.