Self-generated interior blow-up solutions in fractional elliptic equation with absorption

Abstract

In this paper we study positive solutions to problem involving the fractional Laplacian $(E)$ $(-\Delta {)}^{\alpha }u(x)+|u{|}^{p-1}u(x)=0inx\in \Omega \setminus \mathcal{C}$, subject to the conditions $u(x)=0$ $x\in {\Omega }^c$ and ${\lim }_{x\in \Omega \setminus \mathcal{C},x\to \mathcal{C}}u(x)=+\infty$, where $p>1$ and $\Omega$ is an open bounded $C^2$ domain in ${\mathbb{R}}^N$, $\mathcal{C}\subset \Omega$ is a compact $C^2$ manifold with $N-1$ multiples dimensions and without boundary, the operator $(-\Delta {)}^{\alpha }$ with $\alpha \in (0,1)$ is the fractional Laplacian. We consider the existence of positive solutions for problem $(E)$. Moreover, we further analyze uniqueness, asymptotic behaviour and nonexistence.

24 pages