Fractional elliptic problem involving a singularity, a critical exponent and a Radon measure

Abstract

In this paper, we prove the existence of a positive 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 bounded Radon measure in \(\Omega\).