Archive for Rational Mechanics and Analysis

, Volume 165, Issue 4, pp 295-316

First online:

Standing Waves with a Critical Frequency for Nonlinear Schrödinger Equations

  • JAEYOUNG BYEONAffiliated withDepartment of Mathematics POSTECH Pohang, Kyungbuk 790-784 Republic of Korea
  • , ZHI-QIANG WANGAffiliated withDepartment of Mathematics and Statistics Utah State University, Logan, UT 84322

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 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.