Existence of positive solutions for a singular elliptic problem with critical exponent and measure data

Abstract

We prove the existence of a positive {\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \begin{equation} \begin{split} (-\Delta)^su&= \frac{1}{u^\gamma}+\lambda u^{2_s^*-1}+\mu ~\text{in}~\Omega, u&>0~\text{in}~\Omega, u&= 0~\text{in}~\mathbb{R}^N\setminus\Omega. \end{split} \end{equation} Here, $\Omega$ is a bounded domain of $\mathbb{R}^N$, $s\in (0,1)$, $2s<N$, $\lambda,\gamma\in (0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $\mu$ is a nonnegative bounded Radon measure in $\Omega$.