Nonlinear Schrödinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrodinger equation with combined power-type nonlinearities and harmonic potential. By constructing a variational problem and the so-called invariant manifolds of the evolution flow, we derive a sharp condition for blow up and global existence of the solution by applying the potential well theory and the concavity method.