Asymptotic profile of a radially symmetric solution with transition layers for an unbalanced bistable equation

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 $xin [0,1]$ and $a'(0)=0$. In particular we consider the case $a(r)=0$ on some interval $Isubset [0,1]$. 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 $varepsilono 0$. We use the variational procedure used in [4] with a few modifications prompted by the presence of the function $h$.