Existence of solution for a system involving fractional Laplacians and a Radon measure

Abstract

An existence of a nontrivial solution in some `weaker' sense of the following system of equations \begin{align*} (-\Delta)^{s}u+l(x)\phi u+w(x)|u|^{k-1}u&=\mu~\text{in}~\Omega\nonumber\\ (-\Delta)^{s}\phi&= l(x)u^2~\text{in}~\Omega\nonumber\\ u=\phi&=0 ~\text{in}~\mathbb{R}^N\setminus\Omega \end{align*} has been proved. Here $s \in (0,1)$, $l,w$ are bounded nonnegative functions in $\Omega$, $\mu$ is a Radon measure and $k > 1$ belongs to a certain range.