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.