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The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e iωt φω(x) for a nonlinear Schrodinger equation with an inhomogeneous nonlinearity V (x)|u| p1 u ,w hereV (x) is proportional to the electron density. Here, ω> 0a ndφω(x) is a ground state of the stationary problem. When V (x) behaves like |x| b at infinity, where 0 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.

Instability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities