A critical fractional choquard problem involving a singular nonlinearity and a radon measure

Abstract

This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential.

$$\begin{aligned} \begin{aligned} (-\Delta )^su-\alpha \frac{u}{|x|^{2s}}&=\lambda u+ u^{-\gamma }+\beta \left( \int _{\Omega }\frac{u^{2_b^*}(y)}{|x-y|^b}dy\right) u^{2_b^*-1}+\mu ~\text {in}~\Omega ,\\ u&>0~\text {in}~\Omega ,\\ u&= 0~\text {in}~\mathbb {R}^N{\setminus }\Omega . \end{aligned} \end{aligned}$$

(0.1)

Here, $\Omega$ is a bounded domain of $\mathbb {R}^N$, $s\in (0,1)$, $\alpha ,\lambda$ and $\beta$ are positive real parameters, $N>2s$, $\gamma \in (0,1)$, $0<b<\min \{N,4s\}$, $2_b^*=\frac{2N-b}{N-2s}$ is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and $\mu$ is a bounded Radon measure in $\Omega$.