Stability of solitary waves for nonlinear Schrِdinger equations with inhomogeneous nonlinearities

We show that the ground-state solitary waves of the critical nonlinear Schrödinger equation iψt(t,r)+Δψ+V(ϵr)|ψ|4/dψ=0 in dimension d≥2 are orbitally stable as ϵ→0 if V(0)V(4)(0)<Gd[V″(0)]2, where Gd is a constant that depends only on d.