Archive for Rational Mechanics and Analysis

, Volume 165, Issue 4, pp 295–316

Standing Waves with a Critical Frequency for Nonlinear Schrödinger Equations

  • JAEYOUNG BYEON
  • ZHI-QIANG WANG

DOI: 10.1007/s00205-002-0225-6

Cite this article as:
BYEON, J. & WANG, Z. Arch. Rational Mech. Anal. (2002) 165: 295. doi:10.1007/s00205-002-0225-6

Abstract

 This paper is concerned with the existence and qualitative property of standing wave solutions \(\) for the nonlinear Schrödinger equation \(\) with E being a critical frequency in the sense that \(\). 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 \(\). 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 \(\) which has been extensively studied in recent years.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • JAEYOUNG BYEON
    • 1
  • ZHI-QIANG WANG
    • 2
  1. 1.Department of Mathematics POSTECH Pohang, Kyungbuk 790-784 Republic of Korea jbyeon@postech.ac.krKR
  2. 2.Department of Mathematics and Statistics Utah State University, Logan, UT 84322 wang@math.usu.edu