2009, 2009(Special): 828-837. doi: 10.3934/proc.2009.2009.828

General uniqueness results and examples for blow-up solutions of elliptic equations

1. 

Department of Mathematics and Computer Science, Virginia State University, Petersburg, Virginia 23806

Received  June 2008 Revised  July 2009 Published  September 2009

In this paper, we establish the blow-up rate of the large positive solution of the singular boundary value problem

$-\Delta u=\lambda u-b(x)uf(u)$       in $ \Omega,$
$u =+\infty$           on $\partial \Omega,$

where $\Omega$ is a ball domain and $b$ is a radially symmetric function on the domain, $f(u)\in C^1[0,\infty)$ satisfies $f(0)=0,$ $f^' (u)>0$ for all $u>0$, and $f(u)$~$F u^{p-1}$ for sufficiently large $u$ with $F>0$ and $p>1$. Naturally, the blow-up rate of the problem equals its blow-up rate for the very special, but important case, when $f(u)=F u^{p-1}$. Some examples are given to illustrate how the blow-up rate depends on the asymptotic behavior of $b$ near the boundary. $b$ can decay to zero as a polynomial, an exponential function, or a function which is not monotone near the boundary.
Citation: Zhifu Xie. General uniqueness results and examples for blow-up solutions of elliptic equations. Conference Publications, 2009, 2009 (Special) : 828-837. doi: 10.3934/proc.2009.2009.828
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