On the multiplicity of solutions of two nonlocal variational problems

We study two nonlocal variational problems in this paper. One models microp hase separation of diblock copolymers and the other models solid-solid phas e transformations that lead to ne structures. We study a parameter range wh ere the problems can be approximated by their asymptotic limits. We nd all the local minimum solutions of the limiting problems. Because these local m inima are isolated, and hence stable under perturbation, near them there ex ist local minimum solutions of the original problems.