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} (-Δ)^su&= \frac{1}{u^γ}+λu^{2_s^*-1}+μ~\text{in}~Ω, u&>0~\text{in}~Ω, u&= 0~\text{in}~\mathbb{R}^N\setminusΩ. \end{split} \end{equation} Here, $\Omega$ is a bounded domain of ${\mathbb{R}}^N$, $s\in (0,1)$, $2s<N$, $\lambda ,\gamma \in (0,1)$, $2_s^{\ast }=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $\mu$ is a nonnegative bounded Radon measure in $\Omega$.