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AIMS Mathematics

2018, Volume 3, Issue 4: 464-484. doi: 10.3934/Math.2018.4.464
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Research article

A semilnear singular problem for the fractional laplacian

  • Facultad de Matematica, Astronomia y Fisica, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina
  • Received: 16 October 2018 Accepted: 22 October 2018 Published: 24 October 2018

  • MSC : Primary 35A15; Secondary 35S15, 47G20, 46E35

  • We study the problem $\left(-\Delta\right) ^{s}u = -au^{-\gamma}+\lambda h$ in $\Omega, $ $u = 0$ in $\mathbb{R}^{n}\setminus\Omega, $ $u>0$ in $\Omega, $ where $0 < s < 1, $ $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{1, 1}$ boundary, $a$ and $h$ are nonnegative bounded functions, $h\not \equiv 0, $ and $\lambda>0.$ We prove that if $\gamma\in\left(0, s\right) $ then, for $\lambda$ positive and large enough, there exists a weak solution such that $c_{1}d_{\Omega}^{s}\leq u\leq c_{2}d_{\Omega}^{s}$ in $\Omega$ for some positive constants $c_{1}$ and $c_{2}.$ A somewhat more general result is also given.

    Citation: Tomas Godoy. A semilnear singular problem for the fractional laplacian[J]. AIMS Mathematics, 2018, 3(4): 464-484. doi: 10.3934/Math.2018.4.464

    Related Papers:

  • We study the problem $\left(-\Delta\right) ^{s}u = -au^{-\gamma}+\lambda h$ in $\Omega, $ $u = 0$ in $\mathbb{R}^{n}\setminus\Omega, $ $u>0$ in $\Omega, $ where $0 < s < 1, $ $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^{1, 1}$ boundary, $a$ and $h$ are nonnegative bounded functions, $h\not \equiv 0, $ and $\lambda>0.$ We prove that if $\gamma\in\left(0, s\right) $ then, for $\lambda$ positive and large enough, there exists a weak solution such that $c_{1}d_{\Omega}^{s}\leq u\leq c_{2}d_{\Omega}^{s}$ in $\Omega$ for some positive constants $c_{1}$ and $c_{2}.$ A somewhat more general result is also given.


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