几类非线性椭圆方程（组）的刘维尔型定理及应用

The Liouville-type theorems are theorems about the non-existence of nontrivial solutions for (systems of) partial differential equations, and they have many important theoretical applications. In this research project, we will mainly study the non-existence of the positive solutions for Lane-Emden equations with weight functions, for Lane-Emden systems and for higher-order Hénon-Lane-Emden systems under some suitable conditions. The key ingredients for our research are to overcome the diffuclty yielded by weight functions, nonlinear coupling terms and higher-order partial differential terms. By introducing the equivalent differential equations for the Lane-Emden equations with weight functions and studying the monotonicity for their positive solutions, establishing some new uniform estimates for the positive solutions of the Lane-Emden systems, and studying the regularity of the positive solutions for higher-order Hénon-Lane-Emden systems at the origin and the nonexisence of the positive solutions for their equivalent integral systems, we will obtain new Liouville-type theorems for these equations and systems. These Liouville-type theorems will play a very important role in our future study of a priori estimates, singularity estimates and the existence of positive solutions for (systems of) nonlinear elliptic equations. Besides, we will also investigate the symmetry properties of the singular solutions for Lane-Emden equations with weight functions, and classify the singular solutions.

刘维尔型定理是关于偏微分方程（组）不存在非平凡解的定理，它们有很多重要的理论应用。本项目中，我们将主要研究带权Lane-Emden方程、Lane-Emden系统和高阶Hénon-Lane-Emden系统的正解在某些条件下的非存在性问题。重点是克服权函数、非线性耦合项以及高阶偏导数项对我们的研究带来的困难。通过引入与带权Lane-Emden方程等价的微分方程并研究其正解的单调性，对Lane-Emden系统的正解给出新的一致性估计，以及研究高阶Hénon-Lane-Emden系统的正解在原点处的光滑性和与其等价的积分系统正解的非存在性，我们将分别得到新的刘维尔型定理。这些刘维尔型定理对今后我们研究非线性椭圆方程（组）正解的先验估计、奇性估计及正解的存在性具有非常重要的作用。此外，我们也将研究带权Lane-Emden方程奇异正解的对称性和分类。

刘维尔型定理是关于偏微分方程不存在非平凡解的定理，它们有很多重要的理论应用。本项目中，我们主要研究了和Lane-Emden方程相关的几类方程（组）的正解在某些条件下的非存在性问题。重点是克服权函数、非线性耦合项以及高阶偏导数项对我们的研究带来的困难。通过Rellich-Pohozaev型恒等式和对Lane-Emden系统的正解给出新的一致性估计，研究高阶Hénon-Lane-Emden系统的正解在原点处的光滑性和与其等价的积分系统正解的非存在性，以及利用嵌入定理、椭圆理论、Rellich-Pohozaev型恒等式、feedback技巧和解的伸缩不变性研究推广的Hénon-Lane-Emden系统，我们分别对Lane-Emden系统、高阶Hénon-Lane-Emden系统和推广的Hénon-Lane-Emden系统给出新的刘维尔型定理。利用这些刘维尔型定理，我们研究了带权Lane-Emden方程奇异正解的渐近对称性和分类。此外，我们也研究了Wiener–Hopf方程解的渐近展开。