## 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

Zbl 0613.35076
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### References:

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