Standing waves with a critical frequency for nonlinear Schr�dinger equations, II

Jaeyoung Byeon; Zhi-Qiang Wang
October 2003
Calculus of Variations & Partial Differential Equations;Oct2003, Vol. 18 Issue 2, p207
Academic Journal
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 Schr�dinger equations. Our method allows a fairly general class of nonlinearity f( u) including ones without any growth restrictions at large.


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