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

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}$$,subjecttotheconditions$u(x)=0$ $x\in {\mathrm{\Omega }}^c$ and $\lim_{x\in\Omega\setminus$\mathcal{C}$, x\to$\mathcal{C}$}u(x)=+\infty$,where$p>1$and$\Omega$isanopenbounded$C^2$domainin$$\mathbb{R}$^N$,$$\mathcal{C}$\subset \Omega$isacompact$C^2$manifoldwith$N-1$multiplesdimensionsandwithoutboundary,theoperator$(-\Delta)^{\alpha}$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.