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.
Categories: math.AP