Standing waves with a critical frequency for nonlinear Schrodinger equations

This paper is concerned with the existence and qualitative property of standing wave solutions psi(t, x) = e(-iEt/h)v(x) for the nonlinear Schrodinger equation (h) over bar +(h) over bar (2)/2 Deltapsi - V(x)psi + \psi\(p-1) psi = 0 with E being a critical frequency in the sense that min(RN) V(X) = E. We show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of V and whose amplitude goes to 0 as (h) over bar --> 0. Moreover, depending upon the local behaviour of the potential function V (x) near the minimum points, the limiting profile of the standing-wave solutions will be shown to exhibit quite different characteristic features. This is in striking contrast with the non-critical frequency case (inf(RN) V (X) > E) which has been extensively studied in recent years.