Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 05, pp. 1-12.

Title: Asymptotic profile of a radially symmetric solution
       with transition layers for an unbalanced bistable equation
       
Author: Hiroshi Matsuzawa (Numazu National College of Tech., Japan)

Abstract: 
 In this article, we consider the semilinear elliptic problem
 $$
 -\varepsilon^{2}\Delta u=h(|x|)^2(u-a(|x|))(1-u^2)
 $$
 in $B_1(0)$ with the Neumann boundary condition.
 The function $a$ is a $C^1$ function satisfying $|a(x)|< 1$
 for $x\in [0,1]$ and $a'(0)=0$. In particular we consider
 the case $a(r)=0$ on some interval $I\subset [0,1]$.
 The function $h$ is a positive $C^1$ function satisfying $h'(0)=0$.
 We investigate an asymptotic profile of the global minimizer
 corresponding to the energy functional as $\varepsilon\to 0$.
 We use the variational procedure used in [4] with a few
 modifications prompted by the presence of the function $h$.
 
Submitted August 31, 2005. Published January 11, 2006.
Math Subject Classifications: 35B40, 35J25, 35J55, 35J50, 35K57.
Key Words: Transition layer; Allen-Cahn equation; 
           bistable equation; unbalanced