Coexistence phenomenon of concentration and transition of an inhomogeneous phase transition model on surfaces

Abstract

We consider the following singularly perturbed elliptic problemε2Δ ũ + (ũ –a(ŷ))(1- ũ2)=0 in M&ytilde;where M is a two dimensional smooth compact Riemannian manifold associated withmetric ğ, ε is a small parameter.The inhomogeneous term -1 ’ = { ŷ ∈ M : a(ŷ) = 0} is a closed, smooth curve thatΓ’ separate M into two disjoint components M+ and M-and also ∂a/∂v’ > 0 on Γ’,where v’ is the normal of Γ’ pointing to the interior of M-.Moreover the maximum value loop Γ = { ŷ ∈ M : a(ŷ) = b} isa closed, smooth geodesic contained in M in such a way and Γ separate M-into two disjoint components.We will show the existence of solution possessing both transition and concentration phenomenon, i.e.uε → + 1 in M-\ Γδ, uε → -1 in M+, uε → 1 – C along Γ as ε → 0,where Γδ is a small neighborhood of Γ and C is a fixed positive constant.