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On the bound states of the nonlinear Schrödinger equation with a linear potential. (English) Zbl 0694.35202

Summary: We study the nonlinear Schrödinger equation with an “attractive” linear potential: \[ i\phi_ t=-\Delta \phi +(V(x)-| \phi |^{2\sigma})\phi,\quad 0<\sigma <2/(n-2)\quad (NLS), \] which arises in the mathematical description of phenomena in nonlinear optics and plasma physics.
Nonlinear bound states are finite energy localized solutions which, if dynamically stable, play an important role in the structure of general solutions of NLS. We discuss the existence and nonlinear orbital stability of nonlinear ground states of NLS. In particular, if \(-\Delta +V\) supports a linear bound state, then NLS has stable nonlinear ground states in the supercritical and critical cases (\(\sigma\geq 2/n)\), where blow-up (self-focusing or collapse) can occur. This is a phenomenon not present in the case where \(V\equiv 0\). In addition, island of stability in the regime of large \(H^ 1\) norm exist.

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35A15 Variational methods applied to PDEs
35J10 Schrödinger operator, Schrödinger equation
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