On interacting bumps of semi-classical states of nonlinear Schrödinger equations

Abstract

We study concentrated positive bound states of the following nonlinear Schrödinger equation: h2Δu - V (x)u + up = 0, u>0, x∈ RN, where p is subcritical. We prove that, at a local maximum point x0 of the potential function V (x) and for arbitrary positive integer K(K > 1), there always exist solutions with K interacting bumps concentrating near x0. We also prove that at a nondegenerate local minimum point of V (x) such solutions do not exist.