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

Pages: 828 - 837, Issue Special, September 2009

 Abstract        Full Text (184.6K)              

Zhifu Xie - Department of Mathematics and Computer Science, Virginia State University, Petersburg, Virginia 23806, United States (email)

Abstract: 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.

Keywords:  Elliptic equation, uniqueness, blow-up rate, large positive solution, sub solution, super solution
Mathematics Subject Classification:  Primary: 35J25, 35J65, 35J67}% Secondary: 53C35

Received: June 2008;      Revised: July 2009;      Published: September 2009.