On the Stationary Cahn–Hilliard Equation: Interior Spike Solutions

We study solutions of the stationary Cahn–Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the “nondegenerate peak” points) have the following property: There exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak convergence of measures, and Liapunov–Schmidt reduction.