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

Abstract

We prove the existence of a positive (Solutions Obtained as Limits of Approximations) to the following PDE involving fractional power of Laplacian

$${\left(-\mathrm{\Delta }\right)}^{s}u=\frac{1}{u^{\gamma }}+\lambda u^{2_s^{\ast }-1}+\mu \text{in }\mathrm{\Omega },u>0\text{in }\mathrm{\Omega },u=0\text{in }{ℝ}^N\setminus \mathrm{\Omega }.$$

Here, $\mathrm{\Omega }$ is a bounded domain of ${ℝ}^N$, $s\in \left(0,1\right)$, $2s<N$, $\lambda ,\gamma \in \left(0,1\right)$, $2_s^{\ast }=2N∕N-2s$ is the fractional critical Sobolev exponent and $\mu$ is a nonnegative bounded Radon measure in $\mathrm{\Omega }$.