Standing Waves with a Critical Frequency for Nonlinear Schrödinger Equations

@inproceedings{Byeon2002StandingWW,
  title={Standing Waves with a Critical Frequency for Nonlinear Schr{\"o}dinger Equations},
  author={Jaeyoung Byeon and Zhi-qiang Wang},
  year={2002}
}
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… CONTINUE READING

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