非线性Kirchhoff方程与二阶脉冲Hamilton系统相关问题的研究

Based on the actual background of free vibration for elastic string and Celestial mechanics, the research of solution to Kirchhoff equations and the second order impulsive Hamilton systems has important practical significance and theoretical value. The study has been widely concerned by domestic and foreign counterparts. By constructing proper manifold, this project will first consider the existence and the non existence of solutions for a class of Kirchhoff equations when the nonlinear term does not meet certain monotonicity conditions. Next, by using penalization method, the prior estimate of elliptic equation and the energy analysis, we will study the existence of multi-peak solutions and the gradual behavior of the solutions for a class of Kirchhoff equations. So, the classic results for Schrodinger equation will be extended to the nonlocal elliptic problems. Finally, the multiplicities of periodic solutions for the second order impulsive Hamilton systems on time scales are considered under the conditions of where the potential changes sign and without any symmetry on potential function. In particular, we will get estimation of lower bounds for the number of periodic solutions. Furthermore, we will accurately depict the relationship between estimation of lower bounds and the number of pulse frequency in one cycle of periodic solution, and highlight the pulse's dominance. The issues discussed in this project related to mathematics, mechanics, physics and so on. So, the results obtained in this project will be helpful to study related problems for these disciplines.

源于弹性弦的自由振动和天体力学的实际背景，有关非线性Kirchhoff方程和二阶脉冲Hamilton系统解的研究具有非常重要的实际意义和理论价值，受到了国内外同行的广泛关注。本项目首先构造恰当的流形，在非线性项不满足某种单调性条件下，考察一类非线性Kirchhoff方程解的存在性和非存在性。其次，应用惩罚方法结合椭圆方程的先验估计和能量分析，研究一类非线性Kirchhoff方程多峰解的存在性及解的渐近行为，从而把针对Schrodinger方程的经典结果推广到非局部椭圆问题。最后，在位势函数不具有任何对称性或可变号条件下，研究时标上二阶脉冲Hamilton系统周期解的多重性。特别地，得到脉冲系统周期解个数的下界估计以及精确刻划下界估计与系统在周期解的一个周期内发生脉冲次数之间的关系，以凸显脉冲的主导地位。本项目讨论的问题涉及数学、力学、物理等学科，故所得成果将有助于这些学科相关问题的研究。

源于物理模型、生物现象及天体力学的实际背景，有关非线性Kirchhoff方程、Schrödinger–Poisson系统、分数阶薛定谔方程和二阶Hamilton系统解的研究具有非常重要的实际意义和理论价值，受到了国内外同行的广泛关注。本项目首先通过在变号Nehari流形上的极小化讨论，考察了几类非线性Kirchhoff方程和Schrödinger–Poisson系统变号解的存在性，同时对解的渐近行为和其形态进行分析研究；特别地，在临界增长条件下对Kirchhoff方程和Schrödinger–Poisson系统的变号解相关问题进行了讨论。其次，在势与非线性项满足渐近周期条件下，应用变分方法分别研究了带次临界或临界指标的Schrödinger–Poisson系统正或非负基态解的存在性。此外，在位势函数不满足标准的周期性条件或线性增长条件或可变号条件下，研究了几类二阶（或离散）Hamilton系统周期解的存在性和多重性。最后，对几类分数阶薛定谔方程和一类拟线性薛定谔方程非平凡解的存在性及多重性做了一些探讨。 本项目讨论的问题涉及数学、力学、物理等学科，故所得成果将有助于这些学科相关问题的研究。