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

Abstract

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