摘要
本文考虑一类带调和势的非线性 Schrdinger 方程 it=-△+|x|~2-μ||^(p-1)-λ||^(q-1),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<(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?
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?
出处
《数学年刊(A辑)》
CSCD
北大核心
2006年第2期231-238,共8页
Chinese Annals of Mathematics
基金
国家自然科学基金(No.10271084)资助的项目