Existence of positive weak solutions for a nonlocal singular elliptic system

Tomas Godoy; Tomas Godoy

doi:10.3934/math.2019.3.792

AIMS Mathematics

2019, Volume 4, Issue 3: 792-804. doi: 10.3934/math.2019.3.792

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Research article

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

Received: 18 April 2019 Accepted: 27 June 2019 Published: 05 July 2019
MSC : Primary 35A15; Secondary 35S15, 47G20, 46E35

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 give sufficient 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 functions such that $\inf_{\Omega}a>0$ and $\inf_{\Omega}b>0.$ For the found weak solution $\left(u, v\right), $ the behavior of $u$ and $v$ near $\partial\Omega$ is also investigated.
- 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[J]. AIMS Mathematics, 2019, 4(3): 792-804. doi: 10.3934/math.2019.3.792

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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 give sufficient 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 functions such that $\inf_{\Omega}a>0$ and $\inf_{\Omega}b>0.$ For the found weak solution $\left(u, v\right), $ the behavior of $u$ and $v$ near $\partial\Omega$ is also investigated.

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