Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities

The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e ^{i ω t} φ_ω(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 φ_ω(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 ^{i ω t} φ _ω(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.