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).


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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