Interior layers for an inhomogeneous Allen–Cahn equation

We consider the problem where Ω is a smooth and bounded domain in R2 and V is a positive smooth function in . Let Γ be a closed, non-degenerate geodesic with respect to the metric in Ω. We prove that there exist two interior transition layer solutions when ε is sufficiently small. One of the layer solutions approaches −1 in Ω1 and +1 in as ε tends to 0, while the other solution exhibits a transition layer in the opposite direction.