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 ei ω tϕω(x) for a nonlinear Schrödinger equation with an inhomogeneous nonlinearity V (x)|u|p − 1u, 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 ei ω tϕω(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.