Solutions with transition layer and spike in an inhomogeneous phase transition model

We consider the following Singularly perturbed elliptic problem epsilon 2 Delta(u) over bar + ((u) over bar - a((y) over bar)(1- (u) over bar (2)) = 0 in Omega, partial derivative(u) over bar/partial derivative n=0 on partial derivative Omega. where Omega is a bounded domain in R-2 with smooth boundary, -1 < a(<(y)over bar>) < 1. epsilon is a small parameter, n denotes the outward normal of partial derivative Omega. Assume that Gamma = {<(y)over bar> is an element of Omega: a((y) over bar) = 0} is a simple closed and smooth curve contained in Omega in such a way that Gamma separates Omega into two disjoint components Omega(+) = {(y) over bar is an element of Omega: a((y) over bar) > 0} and Omega(-) = {(y) over bar is an element of Omega: a((y) over bar) < 0} and partial derivative a/partial derivative v > 0 on Gamma, where v is the Outer normal to Omega(--) We will show the existence of a solution up with a transition layer near Gamma and a downward spike near the maximum points of a((y) over bar) whose profile looks like