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

Abstract

In this article, we consider the semilinear elliptic problem -ɛ2 Δu = h(|x|)2 (u - α(|x|)) (1 - u2) in B1(0) with the Neumann boundary condition. The function α is a C1 function satisfying |α(x)| < 1 for x ∈ [0, 1] and α′(0) = 0. In particular we consider the case α(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 ɛ → 0. We use the variational procedure used in [4] with a few modifications prompted by the presence of the function h.