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Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials. (English) Zbl 0693.35132

This paper studies the Lyapunov stability of some semiclassical bound states found by A. Floer and A. Weinstein [J. Funct. Anal. 69, 397-408 (1986; Zbl 0613.35076)] and by the author [Commun. Partial Differ. Equations 13, No.12, 1499-1519 (1988)] of the nonlinear Schrödinger equation \[ i\hslash \partial \psi /\partial t=-(\hslash^ 2/2)\Delta \psi +V\psi -| \psi |^{p-1}\psi,\quad 1\leq p<1+4/n. \] It proves that among the bound states, those which are concentrated near local minima (respectively maxima) of the potential V are stable (respectively unstable). It also proves that those bound states are positive if \(\hslash >0\) is sufficiently small.
More results on multi-lump bound states have been obtained also by the author [Commun. Math. Phys. (in press)].
Reviewer: Y.G.Oh

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35K55 Nonlinear parabolic equations

Citations:

Zbl 0613.35076
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References:

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