Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 147, pp. 1-18.
Title: Uniqueness and asymptotic behavior of boundary blow-up solutions
to semilinear elliptic problems with non-standard growth
Authors: Shuibo Huang (Lanzhou Univ., Gansu, China)
Wan-Tong Li (Lanzhou Univ., Gansu, China)
Qiaoyu Tian (Gansu Normal Univ. China)
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.