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 $(E)$ $(-\Delta)^{\alpha} u(x)+|u|^{p-1}u(x)=0 in x\in\Omega\setminus\mathcal{C}$, 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.