Existence of positive weak solutions for a nonlocal singular elliptic system

Tomas Godoy

doi:10.3934/math.2019.3.792

AIMS Mathematics, 2019, 4(3): 792-804. doi: 10.3934/math.2019.3.792. Research article

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Existence of positive weak solutions for a nonlocal singular elliptic system

Tomas Godoy

Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina

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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.

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Keywords: fractional singular elliptic systems; positive solutions; sub and supersolutions; Schauder fixed point theorem

Citation: Tomas Godoy. Existence of positive weak solutions for a nonlocal singular elliptic system. AIMS Mathematics, 2019, 4(3): 792-804. doi: 10.3934/math.2019.3.792

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