Stationary solutions for the Cahn-Hilliard equation

We study the Cahn-Hilliard equation in a bounded domain without any symmetry assumptions. We assume that the mean curvature of the boundary has a nondegenerate critical point. Then we show that there exists a spike-like stationary solution whose global maximum lies on the boundary. Our method is based on Lyapunov-Schmidt reduction and the Brouwer fixed-point theorem.

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