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

TL;DR

This paper investigates the existence and multiplicity of positive solutions to a fractional Lane-Emden elliptic system involving measures, establishing conditions for minimal solutions and multiple solutions with regularity considerations.

Contribution

It introduces new existence and multiplicity results for fractional elliptic systems with measure data, extending previous work to include measure inputs and regularity analysis.

Findings

Existence of minimal positive solutions under small measure mass

Multiple positive solutions when measures are in L^r with r > N/(2s)

Regularity results for solutions based on data conditions

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.