Construction of various types of solutions for an elliptic problem

In this paper, we consider the following elliptic problem... where a(x) is a continuous function satisfying 0 < a(x)<1 for x∈...,Ω is a bounded domain in R^N with smooth boundary, ε > 0 is a small number. A solution of (1.1) can be interpreted as a steady state solution of the corresponding problem: ut=ε²Δu+f(x,u), where f(x,t)=t(t-a(x))(1-t), which arises in a number of places such as population genetics [24,33]. The readers can refer to [22] for more background material for problem (1.1).