Higher Energy Solutions in the Theory of Phase Transitions: A Variational Approach

Abstract

We establish the existence of a higher-energy solution to the vector Ginzburg–Landau equation with a triple-well potential on a bounded and smooth domain on the plane. This solution is obtained by a linking argument. In implementing this variational approach we make several considerations on the dynamics of the negative gradient flow. In particular, we use the Conley index to contruct a suitable one-dimensional invariant set. This solution has Morse index two in the nondegenerate case. We discuss its structure in connection with the so-called triple-junction configurations.