Asymptotic Profile of a Radially Symmetric Solution with Transition Layers for an Unbalanced Bistable Equation

In this article, we consider the semilinear elliptic problem −ε∆u = h(|x|)(u− a(|x|))(1− u) in B1(0) with the Neumann boundary condition. The function a is a C1 function satisfying |a(x)| < 1 for x ∈ [0, 1] and a′(0) = 0. In particular we consider the case a(r) = 0 on some interval I ⊂ [0, 1]. The function h is a positive C1 function satisfying h′(0) = 0. We investigate an asymptotic profile of the global minimizer corresponding to the energy functional as ε → 0. We use the variational procedure used in [4] with a few modifications prompted by the presence of the function h.