## Standing waves with a critical frequency for nonlinear Schrödinger equations. II.(English)Zbl 1073.35199

Summary: For elliptic equations of the form $$\Delta u -V(\varepsilon x) u + f(u) = 0$$, $$x\in \mathbb{R}^N$$, where the potential $$V$$ satisfies $$\liminf_{| x|\to \infty} V(x) > \inf_{\mathbb{R}^N} V(x) =0$$, 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. For Part I, see Arch. Ration. Mech. Anal. 165, No. 4, 295–316 (2002; Zbl 1022.35064).

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35A15 Variational methods applied to PDEs 35J35 Variational methods for higher-order elliptic equations 35J60 Nonlinear elliptic equations

Zbl 1022.35064
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