Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations I

We consider the singularly perturbed equation -e2?u+V(x)u=K(x)up-1 on a domain O?RN which may be bounded or unbounded. Under suitable hypotheses on V , K we construct layered solutions u?H01(O) which concentrate on certain high-dimensional subsets of O . This gives a positive answer to a problem proposed by Ambrosetti, Malchiodi and Ni in [1].