非线性算子方程的单边全局分歧理论及其应用

The studying of nonlinear differential equations is a very hot research topic in the field of nonlinear science. Transforming differential equations into operator equations is an effective method. Conversely, the bifurcation theory of nonlinear operator equations with parameter can be applied to specific nonlinear differential equations. The study on bifurcation phenomena of operator equations with linear compact operator has attracted many famous mathematician's attentions since 1960s. Many celebrated and profound results have been obtained on this topic. However, many nonlinear differential equations can only transform into the operator equations with homogeneous compact operator or fully nonlinear operator. Therefore, the bifurcation theory of operator equations with linear compact operator is not applicable. Therefore, we shall study the unilateral global bifurcation theory of the operator equations with homogeneous compact operator and fully nonlinear operator, respectively. The unilateral global bifurcation results which we shall obtain extend the previous results. Based on the unilateral global bifurcation theorems, we shall investigate the existence, multiplicity, uniqueness and nonexistence of one-sign solutions of Kirchhoff type equations and Monge-Ampere equations. In addition, to study the unilateral global bifurcation phenomenon of the two kinds of equations, we also plan to establish the spectral theory of the corresponding eigenvalu problems. The results we shall obtain not only answer to some open problems, but also improve some relevant results.

非线性微分方程是非线性科学中非常活跃的研究领域。把微分方程转化成算子方程是一个有效的方法。反过来，对带参数非线性算子方程的研究结果可以应用到具体的非线性微分方程中。自上世纪六十年代以来，带线性紧算子的算子方程的分歧现象受到国内外众多知名数学家的关注，并取得了许多深刻的结果。然而，许多非线性微分方程只能转化为带齐次紧算子或完全非线性紧算子的算子方程。因此，带线性紧算子的算子方程的分歧理论就不再适用。本项目计划研究带齐次紧算子和完全非线性紧算子的算子方程的单边全局分歧理论。将要建立的单边全局分歧结果推广了之前的经典结果。也计划借助于将建立的单边全局分歧理论研究Kirchhoff型方程和Monge-Ampere方程定号解的存在性、多解性、惟一性和非存在性。此外，为了研究这两类方程的单边全局分歧现象，也计划建立对应特征值问题的谱理论。将获得的结果既回答了之前的公开问题，也改进了之前的相关结果。

非线性微分方程是非线性科学中非常活跃的研究领域。把微分方程转化成算子方程是一个有效的方法。反过来，对带参数非线性算子方程的研究结果可以应用到具体的非线性微分方程中。自上世纪六十年代以来，带线性紧算子的算子方程的分歧现象受到国内外众多知名数学家的关注，并取得了许多深刻的结果。然而，许多非线性微分方程只能转化为带齐次紧算子或完全非线性紧算子的算子方程。因此，带线性紧算子的算子方程的分歧理论就不再适用。. 本项目获得了带齐次或完全非线性紧算子的算子方程的若干单边全局分歧理论及其在究Kirchhoff型方程和Monge-Ampere方程中的应用，还对几个相关特征值问题建立了谱理论。还建立了几类非线性微分方程定号和变号解的存在性、非存在性、唯一性及多解性结果，肯定回答了美国数学家Junping Shi, Shouchuan Hu和Haiyan Wang等提出的公开问题。还提出了Picone-Hessian恒等式等新概念，形成了分支逼近法等新方法。. 我们按计划完成本项目所有内容，实现了预期目标，解决了关键问题，取得了一系列创新性成果。项目组发表SCI检索论文19篇，主持人为第一或通讯作者的论文17篇，多篇论文发表在《J. Funct. Anal.》、《Calc. Var. Partial Differential Equations》、《J. Differential Equations》、《Commun. Contemp. Math.》、《DiscreteContin. Dyn. Syst.》、《Differential Integral Equations》、《NoDEA Nonlinear Differential Equations Appl.》、《Adv. Nonlinear Stud.》及《Topol. Methods Nonlinear Anal》等国际权威杂志上。部分论文达到国际领先水平，在国内外同行中引起较大影响。主持人所发表的论文被国内外同行引用达473次。主持人在科学出版社出版专著1部。获得省自然科学奖二等奖1次，高校科技进步一等奖3次。. 上述结果和方法具有非常重要的应用价值。特别是在线社交网络信息扩散的规律对网络舆情监控、商业竞争、总统竞选、打击恐怖主义等具有很好的应用前景。