Stability of standing waves for nonlinear Schrodinger equations with inhomogeneous nonlinearities

The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e(i omega t)phi(omega)(x) for a nonlinear Schrodinger equation with an inhomogeneous nonlinearity V (x)| u|(p-1)u, where V (x) is proportional to the electron density. Here, omega > 0 and phi(omega)(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 omega t)phi(omega)(x) is stable for p < 1+(4-2b)/n and sufficiently small omega > 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.