Standing Waves with a Critical Frequency for Nonlinear Schrödinger Equations
- Cite this article as:
- BYEON, J. & WANG, Z. Arch. Rational Mech. Anal. (2002) 165: 295. doi:10.1007/s00205-002-0225-6
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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.