Boundary blow-up solutions to  fractional ellipticequations in a measure framework

Huyuan  Chen; Hichem Hajaiej; Ying  Wang

doi:10.3934/dcds.2016.36.1881

Article Contents

2016, Volume 36, Issue 4: 1881-1903. Doi: 10.3934/dcds.2016.36.1881

This issue Previous Article Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations Next Article On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

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

1.
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022
2.
Department of Mathematics, King Saud University, P.O. Box 2455, 11451 Riyadh

Received: January 2015

Revised: July 2015

Published: May 2017

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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 $$ \begin{array}{lll} \ \ \ (-\Delta)^\alpha u+g(u)=0\quad & {\rm in}\quad \Omega,\\ \phantom{------\ } \ \ \ u=0\quad & {\rm in}\quad \mathbb{R}^N\setminus\bar\Omega,\\ \phantom{} \lim_{x\in\Omega,x\to\partial\Omega}u(x)=+\infty. \end{array} $$

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Mathematics Subject Classification: 35R11, 35J61, 35R06.

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