On variational characterization of four-end solutions of the Allen–Cahn equation in the plane

In this paper a novel variational method is developed to construct four-end solutions in R2R2 for the Allen–Cahn equation. Four-end solutions have been constructed by Del Pino, Kowalczyk, Pacard and Wei when the angle θ   of the ends is close to π/2π/2 or 0 using the Lyapunov–Schmid reduction method, and later by Kowalczyk, Liu and Pacard using a continuation method for general θ∈(0,π/2)θ∈(0,π/2). By a special mountain pass argument in a restricted space, namely a set of monotone paths of monotone functions, a family of solutions in bounded domain is constructed with very good control of their nodal sets, which then leads to a four-end solution after sending the domains to R2R2. In this way, not only a family of four end solutions is constructed for any angle θ∈(0,π/2)θ∈(0,π/2). The Morse index of such a four-end solution is also shown to be one.