Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials (Q908434)

This paper studies the Lyapunov stability of some semiclassical bound states found by \textit{A. Floer} and \textit{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)].