Standing Waves with a Critical Frequency for Nonlinear Schrödinger Equations

Abstract

This paper is concerned with the existence and qualitative property of standing wave solutions ψ(t, x) = e −iEt/h v(x) for the nonlinear Schrödinger equationWe 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 → 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 R N V (x) > E) which has been extensively studied in recent years.

Singular elliptic problems with lack of compactness

“…In view of its proof, we deduce that for all λ ∈ (0, λ 0 ) the conclusion of Lemma 4 holds. Therefore, we obtain the existence of a solution u 0 of problem (2) such that J λ (u 0 ) = c 0,λ .…”

Fractional NLS equations with magnetic field, critical frequency and critical growth

“…It is worth mentioning that the study of fractional Schrödinger equations with the critical frequency was first investigated by Byeon and Wang [4,5]. Main difficulties arise, when dealing with this problem, because of the appearance of the magnetic field and the critical frequency, and of the nonlocal nature of the fractional Laplacian.…”

Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

“…In this sense, E = inf x∈R N W (x) is called a critical frequency (or energy) for the nonlinear Schrödinger equation (1.1), or problem (1.2), in [4] by Byeon and Wang. Under the condition inf x∈R N W (x) > E, there have been enormous investigations on problem (1.2). In [18], Floer and Weinstein considered the case N = 1 and p = 3.…”

“…It seems that Byeon and Wang [4] were the first to study energy level and the asymptotic behavior of positive solutions to problem (1.2) under the condition inf int(A), they proved in [4] that there is a solution u such that…”

“…In [4], Byeon and Wang also proved the existence of solutions concentrating on an isolated zero point b of V (x). Their result shows that the height of the bump of the solution concentrating on b and that of the solution concentrating on Ω are of different order.…”