Boundary blow-up solutions to fractional elliptic equations in a measure framework

Abstract

Let $\alpha\in(0,1)$, $\Omega$ be a bounded open domain in $\mathbb{R}^N$ ($N\ge 2$) 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{equation}\label{0.1}\begin{array}{lll} (-\Delta)^\alpha u+g(u)=k\frac{\partial^\alpha \omega}{\partial \vec{n}^\alpha}\quad & {\rm in}\quad \bar\Omega, (1)\\ \phantom{ (-\Delta)^\alpha +g(u)}u=0\quad & {\rm in}\quad \bar\Omega^c\end{array}\end{equation}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 of (1) is a classical solution of$$ \begin{array}{lll}\ \ \ (-\Delta)^\alpha u+g(u)=0\quad & {\rm in}\quad \Omega,\\\phantom{------\ }\ \ \ u=0\quad & {\rm in}\quad \mathbb{R}^N\setminus\bar\Omega,\\ \phantom{}\lim_{x\in\Omega,x\to\partial\Omega}u(x)=+\infty.\end{array}$$