Standing waves with a critical frequency for nonlinear Schrödinger equations involving critical growth

This paper is concerned with the existence and qualitative property of standing wave solutions ψ(t, x) = e−iEt/h̄v(x) for the nonlinear Schrödinger equation h̄ ∂ψ ∂t + h̄2 2 ψ − V (x)ψ + |ψ |p−1ψ = 0 with E being a critical frequency in the sense that minRN 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 h̄ → 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 (infRN V (x) > E) which has been extensively studied in recent years.