Nonlinear Schroedinger Equation with Inhomogeneous Dirichlet Boundary Data

In this article we study the following nonlinear Schroedinger equation iut=Δu-g vertical bar u vertical bar p-1u in a domain Ω subset of Rn with initial condition u(x,0)=φ(x) and the Dirichlet boundary condition u(x,t)=Q(x,t) on ∂Ω, where φ, Q are given smooth functions. The nonlinear term contributes a negative term to the energy (i.e., g<0). We present the existence theorem for a global solution of finite energy when p≤1+2/n