On the Multiplicity of Solutions of Two Nonlocal Variational Problems

We study two nonlocal variational problems in this paper. One models micro-phase separation of diblock copolymers and the other models solid-solid phase transformations that lead to fine structures. We study a parameter range where the problems can be approximated by their asymptotic limits. We find all the local minimum solutions of the limiting problems. Because these local minima are isolated, and hence stable under perturbation, near them there exist local minimum solutions of the original problems.