Electron. J. Diff. Equ., Vol. 2012 (2012), No. 147, pp. 1-18.

Uniqueness and asymptotic behavior of boundary blow-up solutions to semilinear elliptic problems with non-standard growth

Shuibo Huang, Wan-Tong Li, Qiaoyu Tian

Abstract:
In this article, we analyze uniqueness and asymptotic behavior of boundary blow-up non-negative solutions to the semilinear elliptic equation
$$\displaylines{ \Delta u=b(x)f(u),\quad x\in \Omega,\cr u(x)=\infty, \quad x\in\partial\Omega, }$$
where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, b(x) is a non-negative function on $\Omega$ and f is non-negative on $[0,\infty)$ satisfying some structural conditions. The main novelty of this paper is that uniqueness is established only by imposing a control on their growth on the weights b(x) near $\partial\Omega$ and the nonlinear term f at infinite, rather than requiring them to have a precise asymptotic behavior. Our proof is based on the method of sub and super-solutions and the Safonov iteration technique.

Submitted July 20, 2012. Published August 21, 2012.
Math Subject Classifications: 35J65, 35J60, 74G30, 35B40.
Key Words: Boundary blow-up solutions; uniqueness; asymptotic behavior.

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Shuibo Huang
School of Mathematics and Statistics
Lanzhou University
Lanzhou, Gansu 730000, China
email: huangshuibo2008@163.com
Wan-Tong Li
School of Mathematics and Statistics
Lanzhou University
Lanzhou, Gansu 730000, China
email: wtli@lzu.edu.cn
Qiaoyu Tian
Department of Mathematics
Gansu Normal University for Nationalities
Hezuo, Gansu 747000, China
email: tianqiaoyu2004@163.com

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