Locations of interior transition layers to inhomogeneous transition problems in higher -dimensional domains

Abstract

We consider the following inhomogeneous problems \[ \begin{cases} \epsilon^{2}\mbox{div}(a(x)\nabla u(x))+f(x,u)=0 & \text{ in }\Omega,\\ \frac{\partial u}{\partial \nu}=0 & \text{ on }\partial \Omega,\\ \end{cases} \] where $\Omega$ is a smooth and bounded domain in general dimensional space $\mathbb {R}^{N}$ , $\epsilon >0$ is a small parameter and function $a$ is positive. We respectively obtain the locations of interior transition layers of the solutions of the above transition problems that are $L^{1}$ -local minimizer and global minimizer of the associated energy functional.