Calculus of Variations and Partial Differential Equations

, Volume 18, Issue 2, pp 207–219

Standing waves with a critical frequency for nonlinear Schrödinger equations, II


DOI: 10.1007/s00526-002-0191-8

Cite this article as:
Byeon, J. & Wang, ZQ. Cal Var (2003) 18: 207. doi:10.1007/s00526-002-0191-8


For elliptic equations of the form \(\Delta u -V(\varepsilon x) u + f(u)=0, x\in {\bf R}^N\), where the potential V satisfies \(\liminf_{\vert x\vert\to \infty} V(x) > \inf_{{\bf R}^N} V(x) =0\), we develop a new variational approach to construct localized bound state solutions concentrating at an isolated component of the local minimum of V where the minimum value of V can be positive or zero. These solutions give rise to standing wave solutions having a critical frequency for the corresponding nonlinear Schrödinger equations. Our method allows a fairly general class of nonlinearity f(u) including ones without any growth restrictions at large.

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© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsPOSTECHPohang KyungbukRepublic of Korea
  2. 2.Department of Mathematics and StatisticsUtah State UniversityLoganUSA