Interior layers for an inhomogeneous Allen–Cahn equation

We consider the problemε2Δu+V(y)u(1−u2)=0inΩ,∂u∂n=0on∂Ω where Ω   is a smooth and bounded domain in R2R2 and V   is a positive smooth function in Ω¯. Let Γ   be a closed, non-degenerate geodesic with respect to the metric ds2=V(y)(dy12+dy22) in Ω  . We prove that there exist two interior transition layer solutions uε(1),uε(2) when ε   is sufficiently small. One of the layer solutions uε(1) approaches −1 in Ω1Ω1 and +1 in Ω2=Ω\Ω¯1 as ε   tends to 0, while the other solution uε(2) exhibits a transition layer in the opposite direction.