Global existence and blow up of solutions for the inhomogeneous nonlinear Schrödinger equation in R2R2

This paper discusses a class of inhomogeneous nonlinear Schrödinger equation{i∂tu(t,x)=−Δu(t,x)−V(x)|u(t,x)|p−1u(t,x),u(0,x)=u0(x), where (t,x)∈R×R2(t,x)∈R×R2, V(x)V(x) satisfies some assumptions.By a constrained variational problem, we firstly define some cross-constrained invariant sets for the inhomogeneous nonlinear Schrödinger equation, then we obtain some sharp conditions for global existence and blow up of solutions. As a consequence it is shown that the solution is globally well-posed in Hr1(R2) with the H1H1-norm of the initial data u0u0 which is dominated by the minimal value of the constrained variational problem.