分数阶Yamabe方程及相关非局部问题的研究

在非线性分析的框架下，结合共形几何的方法，发展并完善约化理论，对某些共形协变方程及相关的非局部问题进行研究，具体包括：利用共形几何的方法考虑分数阶Yamabe方程的对称奇异正解的完全分类，同时研究一般奇解附近的解的渐近性质；考虑高阶的双调和的Q-曲率问题，研究对称奇解的分类及渐近性质，同时推广奇解存在性的结论到此方程；研究相关的分数阶Caffarelli-Kohn-Nirenberg不等式的极值解的存在性及对称性质。我们将深入研究约化方法及共形几何的方法在方程中的应用。

By combining the method in conformal geometry and improving the reduction methods in nonlinear functional analysis, we aim to study some conformal invariant partial differential equations and related nonlocal problems. More precisely, firstly, we will use the ideal from conformal geometry to study the classification of positive singular radial solutions to the fractional Yamabe problem, and also study the asymptotic beheviour of the singular solutions around the singular point. Secondly, we will extend the existence results for singular solutions to the fourth order Q curvature problem. Lastly, we will study the existence and symmetry or symmetry breaking property to the extremal solutions for the fractional Caffarelli-Kohn-Nirenberg inequality. We will explore the applicationof idea in conformal geometry and the reduction method to these partial differential equations.