分数阶数曲率方程及其相关问题的研究

The fractional scalar curvature equations are derived from the fractional conformal geometry, which have a wide range of applications in geometric analysis. The applicant and his collaborators have established Harnack inequality, cylindrical symmetry and the exact asymptotic behavior of positive solutions for subcritical Yamabe equations as well as Liouville type theorems of nonnegative solutions for the fractional elliptic equations. This project aims to further study the Liouville type theorem of positive solutions of the fractional scalar curvature system under more weaker indicators; to obtain the Liouville type theorem of positive solutions for the fractional critical scalar curvature equation containing perturbation term and under the more weaker assumptions to perturbation term; to establish the asymptotic behavior of positive solutions near the origin for the fractional general nonlinear term with isolated singularities, and Schoen type Harnack inequalities for the fractional general nonlinear term; to derive the asymptotic behavior of the positive solutions near the origin for the fractional critical scalar curvature equation with isolated singularities as well as the Liouville type theorem of the positive solutions for the fractional critical scalar curvature equation with isolated singularities on the smooth star-shaped domain. The research of this project, which has important theoretical significance and research value, is one of the focus in the field of mordern analysis.

分数阶数曲率方程源于分数阶共形几何，该方程在几何分析中有着广泛的应用。申请人及其合作者已建立了分数阶孤立子奇异次临界Yamabe方程正解的Harnack不等式、柱径向对称性和精确的渐近行为以及分数阶椭圆方程（组）非负解的Liouville型定理。本项目拟进一步研究：分数阶数曲率方程组在较弱指标下正解的Liouville型定理；含扰动项以及在扰动项较弱假设下的分数阶数曲率临界方程正解的Liouville型定理；分数阶一般非线性项孤立子奇异方程的正解在原点附近的渐近行为，以及分数阶一般非线性项的Schoen型Harnack不等式；分数阶孤立子奇异临界数曲率方程正解在原点附近的渐近行为；分数阶孤立子奇异数曲率临界方程在光滑星型域上正解的Liouville型定理。本项目的研究是现代分析领域热点问题之一，具有重要的理论意义和研究价值。

分数阶方程和最优几何不等式在数学很多分支中有广泛应用。本项目主要围绕这两方面展开，并取得如下成果：1.证明了分数阶P方程正解的单调性和唯一性。2.建立了分数阶高阶静态Schrödinger-Hartree-Maxwell方程非负解的超调和性质以及非负解的分类。3.证明了分数阶Poisson核Stein-Weiss不等式和极值函数存在性，以及相应欧拉方程正解的正则性。4.得到了上半空间Hardy-Littlewood-Sobolev不等式对应欧拉方程非负解的分类。本项目的研究是现代分析领域热点问题之一，具有重要的理论意义和研究价值。