On the energy functionals derived from a non-homogeneous p-Laplacian equation: Γ-convergence, local minimizers and stable transition layers

Abstract

In this paper we consider a family of singularly perturbed non-homogeneous p-Laplacian problems ϵpdiv(k(x)|∇u|p−2∇u)+k(x)g(u)=0 in Ω⊂Rn subject to Neumann boundary conditions. We establish the Γ-convergence of the energy functionals associate to this family of problems. As an application, we obtain the existence and profile asymptotic of a family of local minimizers in the one-dimensional case (i.e. Ω=(0,1)). In particular, these minimizers are stable solutions which develop inner transition layer in (0,1).