Stable transition layers in a balanced bistable equation with degeneracy Original Research Article

In this paper we consider the semilinear elliptic problem: −ε2uxx=u(α(x)2−u2) in (0,1) with the Neumann boundary condition. The function α is positive and is allowed to degenerate at where α takes its local minimum. We construct a stable solution which has exactly one layer near the interval where α degenerates.