Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2019; 56(1): 245-251

Online first article June 28, 2018      Printed January 1, 2019

https://doi.org/10.4134/BKMS.b180221

Copyright © The Korean Mathematical Society.

A note on boundary blow-up problem of $\Delta u=u^p$

Seick Kim

Yonsei University

Abstract

Assume that $\Omega$ is a bounded domain in $\mathbb{R}^n$ with $n\ge 2$. We study positive solutions to the problem, $\Delta u=u^p$ in $\Omega$, $u(x)\to\infty$ as $x\to\partial\Omega$, where $p>1$. Such solutions are called boundary blow-up solutions of $\Delta u=u^p$. We show that a boundary blow-up solution exists in any bounded domain if $1

Keywords: blow-up, semi-linear equation, existence, uniqueness

MSC numbers: Primary 35J65; Secondary 35B05

Supported by: The author was partially supported by NRF Grant No. NRF-20151009350.
This paper is based on a presentation by the author at the 2001 AMS sectional meeting, Williamstown, MA, and was supported in
part by NSF Grant No. DMS-9971052.