On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

Abstract

We study positive solutions to the fractional Lane-Emden system \begin{equation*} \tag{S}\label{S} \left\{ \begin{aligned} (-\Delta)^s u &= v^p+\mu \quad &&\text{in } \Omega \\ (-\Delta)^s v &= u^q+\nu \quad &&\text{in } \Omega\\ u = v &= 0 \quad &&\text{in } \Omega^c={\mathbb R}^N \setminus \Omega, \end{aligned} \right. \end{equation*} where $\Omega$ is a $C^2$ bounded domains in ${\mathbb R}^N$, $s\in(0,1)$, $N>2s$, $p>0$, $q>0$ and $\mu,\, \nu$ are positive measures in $\Omega$. We prove the existence of the minimal positive solution of the above system under a smallness condition on the total mass of $\mu$ and $\nu$. Furthermore, if $p,q \in (1,\frac{N+s}{N-s})$ and $0 \leq \mu,\, \nu\in L^r(\Omega)$ for some $r>\frac{N}{2s}$ then we show the existence of at least two positive solutions of the above system. We also discuss the regularity of the solutions.