Bilayer structures are central to amphiphilic polymer systems which possess a phase which wets two immiscible fluids. The amphiphilic component forms thin layers which separate the immiscible phases. When one of the immiscible phases and the amphiphilic material are proportional, and scarce, then the mixture can be modeled as two phase and the bilayer structures as homoclinic connections. We prove the existence of the bilayer structures (homoclinic solutions) for the functionalized Cahn-Hilliard equation, whose equilibriums support these structures. We employ two methods: a functional analytical approach and a variant of Lin's method. The functional analytical approach is based upon a Newton type contraction mapping and it gives the leading order description of the homoclinic connection in terms of a homoclinic connection of a low-order problem. The contraction mapping construction also requires a non-degeneracy condition, which we conjecture is associated to an orbit-flip bifurcation of the homoclinic connection within the higher-order system. Lin's method is an implementation of the Lyapunov-Schmidt method to prove the existence of heteroclinic chains in dynamical systems. Because of the degeneracy of the full problem, we apply Lin's method only to a more restricted parameter regime.
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