中文摘要：

本文考虑一类带调和势的非线性Schro·2dinger方程其中iφt=-△φ＋｜x｜^2φ-μ｜φ｜^p-1φ-λ｜φ｜^q-1φ,x∈R^N,t≥0,其中μ〉0，λ〉0．当N=1，2时，1〈p〈q〈∞；当N≥3时，1〈p〈q〈等N＋2/N-2,运用精巧的变分方法、势井方法和凸方法，得到了方程的整体解和爆破解存在的门槛．进一步回答了：当q〉p〉1＋4/N时，方程的Cauchy问题的初值小到什么程度，其整体解存在？

英文摘要：

This paper is concerned with a class of nonlinear Schro··dinger equations with a harmonic potential iφt=-△φ＋｜x｜^2φ-μ｜φ｜^p-1φ-λ｜φ｜^q-1φ,x∈R^N,t≥0 where μ〉 0, λ〉0, 1〈p〈q〈N＋2/N-2 when N ≥ 3 and 1〈p〈q〈∞ when N= 1, 2. By an intricate variational argument the authors derive out a threshold of blowing up and global existence by applying the potential well argument and the concavity method. Furthermore, they answer the question： How small are the initial data, the global solutions of the Cauchy problem of above equation exist for q 〉 p 〉 1 ＋ 4/N？