Large solutions to elliptic equations involving fractional Laplacian

Abstract

The purpose of this paper is to study boundary blow up solutions for semi-linear fractional elliptic equations of the form

where $p>1$, $\Omega$ is an open bounded $C^2$ domain of ${\mathbb{R}}^N$, $N\ge 2$, the operator $(-\mathrm{\Delta }{)}^{\alpha }$ with $\alpha \in (0,1)$ is the fractional Laplacian and $f:\Omega \to \mathbb{R}$ is a continuous function which satisfies some appropriate conditions. We obtain that problem (0.1) admits a solution with boundary behavior like $d(x{)}^{-\frac{2\alpha }{p-1}}$, when $1+2\alpha <p<1-\frac{2\alpha }{{\tau }_0(\alpha )}$, for some ${\tau }_0(\alpha )\in (-1,0)$, and has infinitely many solutions with boundary behavior like $d(x{)}^{{\tau }_0(\alpha )}$, when $\max \{1-\frac{2\alpha }{{\tau }_0}+\frac{{\tau }_0(\alpha )+1}{{\tau }_0},1\}<p<1-\frac{2\alpha }{{\tau }_0}$. Moreover, we also obtained some uniqueness and non-existence results for problem (0.1).