An existence and stability result for standing waves of nonlinear Schrödinger equations

We consider a nonlinear Schrodinger equation with a nonlinearity of the form V (x)g(u). Assuming that V (x) behaves like |x|−b at infinity and g(s) like |s|p around 0, we prove the existence and orbital stability of travelling waves if 1 < p < 1 + (4− 2b)/N .

[1]  Kazunaga Tanaka,et al.  Uniqueness of positive radial solutions of semilinear elliptic equations in RN and séré's non—degeneracy condition , 1999 .

[2]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[3]  W. Strauss,et al.  Nonlinear bound states outside an insulated sphere , 1994 .

[4]  J. McLeod,et al.  Stability of standing waves for some nonlinear Schrödinger equations , 2003, Differential and Integral Equations.

[5]  J. Shatah,et al.  Stability theory of solitary waves in the presence of symmetry, II☆ , 1990 .

[6]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

[7]  E. Yanagida.  Uniqueness of positive radial solutions of △u+g(r)u+h(r)up=0 in Rn , 1991 .

[8]  W. Ni,et al.  Locating the peaks of least energy solutions to a semilinear Neumann problem , 1993 .

[9]  G. Perelman.  Stability of solitary waves for nonlinear Schrödinger equation , 1996 .

[10]  W. Rother,et al.  Nonlinear scalar field equations , 1992, Differential and Integral Equations.

[11]  Louis Jeanjean,et al.  Local conditions insuring bifurcation from the continuous spectrum , 1999 .

[12]  M. Weinstein.  Nonlinear Schrödinger equations and sharp interpolation estimates , 1983 .

[13]  Michael I. Weinstein,et al.  On the bound states of the nonlinear Schrödinger equation with a linear potential , 1988 .

[14]  P. Lions,et al.  Orbital stability of standing waves for some nonlinear Schrödinger equations , 1982 .

[15]  Reika Fukuizumi.  Stability and instability of standing waves for nonlinear Schrödinger equations , 2003 .

[16]  V. Rich.  Personal communication , 1989, Nature.

[17]  Massimiliano Berti,et al.  Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities , 2002, math/0211310.

[18]  C. Stuart.  Bifurcation in Lp(RN) for a Semilinear Elliptic Equation , 1988 .

[19]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

[20]  I. Iliev,et al.  Stability and instability of solitary waves for one-dimensional singular Schrödinger equations , 1993, Differential and Integral Equations.

[21]  Catherine Sulem,et al.  The nonlinear Schrödinger equation , 2012 .

[22]  A. D. Bouard,et al.  Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities , 2005 .

[23]  Mihai Mariş,et al.  Existence of nonstationary bubbles in higher dimensions , 2002 .