Existence and orbital stability of standing waves for some nonlinear Schrodinger equations, perturbation of a model case

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:246

Article

Genoud, Francois

关键词: Nonlinear Schrodinger equation;    Standing waves;    Bifurcation;    Orbital stability;   

DOI  :  10.1016/j.jde.2008.10.029

来源: Elsevier

【 摘 要 】

The following nonlinear Schrodinger equation is studied i partial derivative(t) w + Delta w + f(x, w) = 0, w = w(t, x): R x R(N) -> C, N >= 3. f is a nonlinearity that can be written in the form f(x, s) = V(x)vertical bar s vertical bar(p-1)s + r(x, s), where V decays at infinity like vertical bar x vertical bar(-b) for some b is an element of (0, 2) and r is a perturbation having the same qualitative behaviour as V(x)vertical bar s vertical bar(p-1)s for small vertical bar s vertical bar. f is possibly singular at the origin 0 G RN. A standing wave is a solution of the form w(t, x) = e(i lambda t) u (x) where lambda > 0 and u : R(N) -> R. For 1 < p < 1 + (4 - 2b)/(N - 2), the existence in H(1)(R(N)) of a C(1)-branch of standing waves parametrized by frequencies lambda in a right neighbourbood of lambda = 0 is proven. These standing waves are shown to be orbitally stable if 1 < p < 1 + (4 - 2b)/N and unstable if 1 + (4 - 2b)/N < p < 1 + (4 - 2b)/(N - 2). (c) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

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