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 $R^N$ ($N\ge 2$) 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 $$⟨\frac{{\partial }^{\alpha }\omega }{\partial {\vec{n}}^{\alpha }},f⟩={\int }_{\partial \Omega }\frac{{\partial }^{\alpha }f(x)}{\partial {\vec{n}}_x^{\alpha }}d\omega (x),f\in C^1(\bar{\Omega }),$$ where ${\vec{n}}_x$ is the unit outward normal vector at point $x\in \partial \Omega$. In this paper, we prove that problem $$\begin{array}{ll}(-\Delta {)}^{\alpha }u+g(u)=k\frac{{\partial }^{\alpha }\omega }{\partial {\vec{n}}^{\alpha }}& \mathrm{in}\bar{\Omega },\\ u=0& \mathrm{in}{\Omega }^c\end{array}$$ 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 $$ \begin{array}{lll} \ \ \ (-Δ)^αu+g(u)=0\quad & {\rm in}\quad Ω,\\[2mm] \phantom{------\} \ u=0\quad & {\rm in}\quad R^N\setminus\barΩ,\\[2mm] \phantom{} \lim_{x\inΩ,x\to\partialΩ}u(x)=+\infty. \end{array} $$

25 pages. arXiv admin note: text overlap with arXiv:1410.2672

References in corpus (1)

• Existence, Non-existence, Uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary