Electron. J. Diff. Eqns., Vol. 2006(2006), No. 64, pp. 1-9.

A boundary blow-up for sub-linear elliptic problems with a nonlinear gradient term

Zhijun Zhang

Abstract:
By a perturbation method and constructing comparison functions, we show the exact asymptotic behaviour of solutions to the semilinear elliptic problem
$$ \Delta u -|\nabla u|^q=b(x)g(u),\quad u>0 \quad\text{in }\Omega, \quad u\big|_{\partial \Omega}=+\infty, $$
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $q\in (1, 2]$, $g \in C[0,\infty)\cap C^1(0, \infty)$, $g(0)=0$, $g$ is increasing on $[0, \infty)$, and $b$ is non-negative non-trivial in $\Omega$, which may be singular or vanishing on the boundary.

Submitted February 5, 2006. Published May 20, 2006.
Math Subject Classifications: 35J60, 35B25, 35B50, 35R05.
Key Words: Semilinear elliptic equations; large solutions; asymptotic behaviour.

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Zhijun Zhang
Department of Mathematics and Information Science
Yantai University, Yantai, Shandong, 264005, China
email: zhangzj@ytu.edu.cn

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