A Variational Approach to Multiple Layers of the Bistable Equation in Long Tubes

Multiple layer solutions of the balanced bistable equation in infinite tubes are constructed via a variational method. I start with a characterization of Palais-Smale sequences which easily gives some global minima in the desired function classes as single layers. Assuming these minima are isolated as critical points, I paste them together to serve as an approximate multiple layer solution. If there were no exact solutions near the approximate one, the negative gradient flow of the energy functional would significantly lower the energy. On the other hand, if the building minima are kept far from each other, the energy of a function near the approximate solution is not much less than that of the approximate solution. This contradiction proves the existence of a solution.