Existence of positive weak solutions for a nonlocal singular elliptic system

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with $C^{1,1}$ boundary,and let $s\in\left( 0,1\right) $ be such that $s<\frac{n}{2}.$ We givesufficient conditions for the existence of a weak solution $\left(u,v\right) \in H^{s}\left( \mathbb{R}^{n}\right) \times H^{s}\left(\mathbb{R}^{n}\right) $ of the nonlocal singular system $\left(-\Delta\right) ^{s}u=ad_{\Omega}^{-\gamma_{1}}v^{-\beta_{1}}$ in $\Omega,$$\left( -\Delta\right) ^{s}v=bd_{\Omega}^{-\gamma_{2}}u^{-\beta_{2}}$ in$\Omega,$ $u=v=0$ in $\mathbb{R}^{n}\setminus\Omega,$ $u>0$ in $\Omega,$ $v>0$in $\Omega,$ \ where $a$ and $b$ are nonnegative bounded measurable functionssuch that $\inf_{\Omega}a>0$ and $\inf_{\Omega}b>0.$ For the found weaksolution $\left( u,v\right) ,$ the behavior of $u$ and $v$ near$\partial\Omega$ is also investigated.

Keywords

• fractional singular elliptic systems
• positive solutions
• sub and supersolutions
• Schauder fixed point theorem