A semilnear singular problem for the fractional laplacian

Abstract

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 $0{\langle}s\langle1,$ $\Omega$ 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.

Keywords

• singular elliptic problems| positive solutions| fractional Laplacian| sub andsupersolutions