非线性微分方程结点解的全局结构

It is current focus of attention for the mathematician at home and abroad to study the solution structure of nonlinear differential equations by using bifurcation theory. There exists a wide range of practical applications in the fields of Physics, Chemistry, and Modern control theory, etc.. The project is aimed to study its applications and the unilateral global bifurcation phenomena for several classes of nonlinear differential equations including the beam equations and the periodic problems etc.. Firstly, we investigate the spectrum structure of several classes of linear differential equations including the beam equations etc. with constant weights and sign-changing weights and several classes of half-linear eigenvalue problems including the beam equations and the periodic problems etc.. Secondly, we shall establish the unilateral global bifurcation theorems and the unilateral global interval bifurcation theorems for several classes of nonlinear differential equations including the beam equations and the periodic problems etc.. Thirdly, we shall investigate the existence and nonexistence of nodal solutions for several classes of nonlinear differential equations including the beam equations and the periodic problems etc.. Finally, we also study the existence and nonexistence of nodal solutions for several classes of half-linear equations including the beam equations and the periodic problems etc..Achievement of the project will be an example to study other bifurcation problem of nonlinear differential equations. Meanwhile, it also further provides theoretical guidance for numerical calculation in the area of practical applications.

利用分歧理论研究非线性微分方程解的结构目前已成为国内外数学家关注的热点，它在物理、化学及现代控制理论等领域中有着广泛的实际应用背景。本项目计划研究梁方程及周期问题等几类非线性微分方程解集的单侧全局分歧和单侧全局区间分歧结构及其应用。首先，计划建立梁方程等几类带定权函数和带变号权函数的线性特征值问题的谱理论，也计划建立梁方程及周期问题等几类半线性和半拟线性特征值问题的谱理论；其次，计划建立适用于研究梁方程及周期问题等几类非线性微分方程的单侧全局分歧定理和单侧全局区间分歧定理；再次，也计划用上述所建立的谱理论和单侧全局分歧定理及单侧全局区间分歧定理研究梁方程及周期问题等几类非线性微分方程解的存在与否问题；最后，还计划研究梁方程及周期问题等几类半线性特征值问题解的存在与否问题。本项目的研究将为研究其它非线性微分方程分歧问题提供参考方法，同时也能为实际应用领域中进一步进行数值计算提供理论指导。

利用全局分歧结构来刻画带参数的非线性微分方程已成为国内外数学家关注的重点问题，这些问题的研究在物理、化学和现代控制理论等领域中有着广泛的实际应用背景。. 本项目中研究了几类非线性微分方程解集的单侧全局分歧结构和单侧全局区间分歧结构及其应用。首先，建立了两端固定支撑梁方程等几类非线性问题的单侧全局分歧定理，结合序列集取极限方法研究了上述几类非线性问题结点解或正解的存在性定理。其次，建立了两端简单支撑梁方程、p-Laplacian方程及周期问题等几类非线性问题单侧全局区间分歧定理，也建立了对应于上述几类问题的半线性和半拟线性特征值问题的谱理论和相应的扰动问题的单侧全局分歧定理，进而结合连通分支列取极限的技巧获得了上述几类带跳跃非线性项问题结点解或定号解的存在性定理。再次，建立了在零点和无穷远处的Monge-Ampere方程的全局区间分歧定理，研究了此类方程凸解的全局结构存在性问题。最后，建立了 -Hessian方程、Kirchhoff-类问题及平均曲率算子方程等几类高维问题的单侧全局区间分歧定理和相应的的半线性和半拟线性特征值问题的谱理论，进而建立了相应与上述几类半线性和半拟线性特征值扰动问题的单侧全局分歧定理和结点解或定号解的存在性定理。. 上述结果和方法丰富了非线性分析和微分方程的理论内容，对于进一步发展分歧理论及其应用具有非常重要的科学意义。