Let $\alpha\in(0,1)$, $\Omega$ be a bounded open domain in $R^N$ ($N\ge 2$) with $C^2$ boundary $\partial\Omega$ and $\omega$ be the Hausdorff measure on $\partial\Omega$. We denote by $\frac{ \partial^\alpha \omega}{ \partial \vec{ n}^\alpha}$ a measure $$\langle\frac{ \partial^\alpha \omega}{ \partial \vec{ n}^\alpha},f\rangle=\int_{ \partial\Omega}\frac{ \partial^\alpha f(x)}{ \partial \vec{ n}_x^\alpha} d\omega(x),\quad f\in C^1(\bar\Omega),$$ where $\vec{ n}_x$ is the unit outward normal vector at point $x\in\partial\Omega$. In this paper, we prove that problem $$ \begin{ array}{ lll} (-\Delta)^\alpha u+g(u)=k\frac{ \partial^\alpha \omega}{ \partial \vec{ n}^\alpha}\quad & { \rm in}\quad \bar\Omega,\\[2mm] \phantom{ (-\Delta)^\alpha +g(u)} u=0\quad & { \rm in}\quad \Omega^c \end{ array} $$ admits a unique weak solution $u_k$ under the hypotheses that $k>0$, $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$ and $g$ is a nondecreasing function satisfying extra conditions. We prove that the weak solution is a classical solution of $$ \begin{ array}{ lll} \ \ \ (-\Delta)^\alpha u+g(u)=0\quad & { \rm in}\quad \Omega,\\[2mm] \phantom{ ------\} \ u=0\quad & { \rm in}\quad R^N\setminus\bar\Omega,\\[2mm] \phantom{ } \lim_{ x\in\Omega,x\to\partial\Omega}u(x)=+\infty. \end{ array} $$
Categories: math.AP