Small solutions to nonlinear Schrödinger equations in the Sobolev spaces

The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrödinger equations is studied. The global well-posedness of the problem is proved in the Sobolev space H^s = H^s(Rⁿ) of fractional order s > n/2 under the following assumptions. (1) Concerning the Cauchy data φ ∈ H^s: ∥φ; L²∥ is relatively small with respect to ∥φ; H^σ∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f: f(u) behaves as a conformal power u^1+4/n near zero and has an arbitrary growth rate at infinity.