论文标题
对于能量关键和超临界非线性的总彼得维斯基方程的积极解决方案
Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities
论文作者
论文摘要
我们认为具有谐波潜力的总壁杆菌方程的正面和空间腐烂的溶液。对于临界情况，存在且仅当频率属于（1,3）在三个维度和（0，d）中的频率时，存在基态。我们对基态的渐近行为进行精确描述，直到不同值的d值。对于能量危机的情况，存在（0，d）中某个频率的单数解。我们计算径向函数类别中奇异溶液的摩尔斯索引，并表明在振荡情况下，摩尔斯指数是无限的，在单调的情况下，非线性功率不够大，在单调的单调情况下等于1或2，而在单调元件中，对于非线性元件，非线性元件中的非线性功率足够大。
We consider positive and spatially decaying solutions to the Gross-Pitaevskii equation with a harmonic potential. For the energy-critical case, there exists a ground state if and only if the frequency belongs to (1,3) in three dimensions and in (0,d) in d dimensions. We give a precise description on asymptotic behaviors of the ground state up to the leading order term for different values of d. For the energy-supercritical case, there exists a singular solution for some frequency in (0,d). We compute the Morse index of the singular solution in the class of radial functions and show that the Morse index is infinite in the oscillatory case, is equal to 1 or 2 in the monotone case for nonlinearity powers not large enough and is equal to 1 in the monotone case for nonlinearity power sufficiently large.
