CLASS OF NON-LINEAR SCHRODINGER EQUATIONS .1. CAUCHY-PROBLEM, GENERAL-CASE

We study the Cauchy problem for a class of nonlinear Schrödinger equations of the form i( du dt) = (-Δ + m)u + f(u) in Rn with n ≥ 2, where m is a real constant and f a complex valued nonlinear function. Under suitable assumptions on f, we prove the existence and uniqueness of global solutions of the initial value problem in the Sobolev space H1(Rn. The assumptions of f include continuous differentiability, the condition f(0) = 0 and suitable power bounds both at zero and at infinity. They cover the case of a single power f(u) = λ | u |p - 1u where 1 ≤ p < (n + 2) (n - 2) if λ ≥ 0 and 1 ≤ p < (n + 4) n if λ ≤ 0. In subsequent papers, we shall treat the scattering theory for the same class of equations, and a number of special cases, in particular the case of the dimension n = 1. © 1979.