Existence and multiplicity of positive solutions for a critical fractional Laplacian equation with singular nonlinearity

In this paper, we consider the following problem \(\begin{aligned} {\left\{ \begin{array}{ll} (-\varDelta )^{s} u=g(x) u^{2_{s}^{*}-1}+\lambda u^{-\gamma }, &{} \text { in } \varOmega , \\ u>0, \text { in } \varOmega , \quad u=0, &{} \text { on } \partial \varOmega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{(-\Delta )}^{s}u=g(x)u^{2_s^{\ast }-1}+\lambda u^{-\gamma },\hfill & \text{in}\Omega ,\hfill \\ u>0,\text{in}\Omega ,u=0,\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right)\end{array}$$ where \(\varOmega \subset {\mathbb {R}}^{N}(N > 2s)\) $\Omega \subset {\mathbb{R}}^N(N>2s)$ is a smooth bounded domain, \(s\in (0,1)\) $s\in (0,1)$ , \(\lambda \) $\lambda$ is a positive constant, \(0<\gamma <1\) $0<\gamma <1$ , \(2_{s}^{*}=\frac{2 N}{N-2s}\) $2_s^{\ast }=\frac{2N}{N-2s}$ and \((-\varDelta )^{s} \) ${(-\Delta )}^s$ is the spectral fractional Laplacian. Based upon the Nehari manifold and using variational method we relate the number of positive solutions to the global maximum of the coefficient of the critical nonlinearity g.