Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

Abstract

Abstract. The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e iωt φω(x) for a nonlinear Schrödinger equation with an inhomogeneous nonlinearity V (x)|u| p−1 u, where V (x) is proportional to the electron density. Here, ω > 0 and φω(x) is a ground state of the stationary problem. When V (x) behaves like |x| −b at infinity, where 0 < b < 2, we show that e iωt φω(x) is stable for p < 1 + (4 − 2b)/n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) = |x| −b . Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method.

Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

“…After an appropriate rescaling of the variables, this result is proved in [13] using Proposition 5.2. Earlier related results were proved using the condition (SC * ), [9], [20]. Vol.…”

A smooth global branch of solutions for a semilinear elliptic equation on $${\mathbb{R}^N}$$

“…4. The smoothness of the branch requires somewhat more restrictive assumptions on V than what is supposed in [7] (see hypothesis (H4) below) and is needed by our approach to the orbital stability of standing waves for (1). Compared to [7], the method we used in [2] yields more precise information about the behaviour of the solutions as λ → 0.…”

“…2, it is natural to seek solutions of (E λ ) in the Sobolev space H 1 (R N ). The existence and orbital stability of standing waves for (1) has been recently studied by several authors [1,2,7]. In [1] and [7], the existence of standing waves is established by variational arguments and their orbital stability is studied in a right neighbourhood of λ = 0.…”

“…The existence and orbital stability of standing waves for (1) has been recently studied by several authors [1,2,7]. In [1] and [7], the existence of standing waves is established by variational arguments and their orbital stability is studied in a right neighbourhood of λ = 0. In [7], the authors proved that, for 1 < p < 1 + 4−2b N , the non-trivial solutions ϕ λ of (E λ ) that they obtain for small λ > 0 converge to 0 ∈ H 1 (R N ) as λ → 0.…”

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

“…The present paper focuses on the situation where the nonlinearity f is given as a perturbation of the nonautonomous, power-like nonlinearity V (x)|w| p−1 w with p > 1. The work on the latter issue was initiated in [3] and has been the subject of several recent papers, see [1,4,6]. (Actually [6] deals with a slightly more general nonlinearity, see Section 1.1 below.)…”