几类与黎曼几何相关的椭圆问题的研究

Liouville-type equations and their elliptic systems have a wide range of background in geometry, physics, chemistry, biography and other fields. The research results play an important role in resolving practical problems. When the curvature function may not bounded from above or below, we need to study the existence of regularity metric. For one hand, transform the problem to the extreme problem with constrained conditions. Then by using variational methods and Moser-Trudinger inequality, we try to obtain the existence results of conformal metric; for the other hand, we analyze the non-existence of conformal metric by using moving planes method. For Q curvature problem prescribed by Liouville-type equations, a priori estimates, the existence, multiple solutions，concentration compactness and bubble behavior are discussed by using critical point theory and elementary analysis methods. For variable coefficient second order Toda system and higher order elliptic equations, partial classification, asymptotic behavior, a priori estimates and generalized K-W identity are also studied. Furthermore, we can use above methods to solve similar singular elliptic problems.

Liouville型方程及其椭圆系统有着广泛的几何、物理、化学、生物等背景，其研究成果对解决实际问题起着十分重要的指导作用.当Liouville型方程的曲率函数上下无界时，需要对正则度量的存在性进行研究,一方面,将该问题转化为约束条件的极值问题，然后用变分法结合Moser-Trudinger不等式获得共形度量的存在性结果; 另一方面运用移动平面方法分析共形度量的非存在性.对于Liouville型方程所描述的Q曲率问题，用临界点理论结合Moser-Trudinger不等式和基本分析方法讨论解的先验估计、存在性、多重性及解的集中紧性、冒泡行为等.对于变系数的二阶Toda系统和高阶椭圆方程的解进行分类、渐近分析和先验估计，推广著名的K–W恒等式. 进一步将上述研究方法用于解决同类型的奇异椭圆问题.

指数型椭圆方程的定性分析是几何分析领域的一个前沿课题，在几何、物理等领域都有应用。本项目主要针对有界区域（全空间）上的次临界或临界指数型椭圆方程以及次临界和临界指数型增长的分数阶p-Laplacian方程以及共振与非共振情形下的Kirchhoff椭圆型方程问题和具有凹非线性项的分数阶p-Laplacian问题，运用极小极大方法，结合Moser-Trudinger不等式、Moser理论等得到了上述方程的存在性结果；通过改进Brezis和Merle的方法获得了一致先验性估计。在方法上具有一定的创新性，整体研究具有一定的系统性，对相关领域的研究具有一定的参考价值。