Semi-classical states for the Choquard equation

We study the nonlocal equation where , , is the Riesz potential and is a small parameter. We show that if the external potential has a local minimum and then for all small the problem has a family of solutions concentrating to the local minimum of provided that: either , or and , or and . Our assumptions on the decay of and admissible range of are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.