非局部椭圆方程组和加权的积分方程组的正解的若干性质

The fractional Laplacian is a particular class of nonlocal operators, which has received considerable attention in partial differential equation and nonlinear analysis during the last years. The classical Hardy-Littlewood-Sobolev (HLS) inequality, which has been a basic inequality in analytics, plays the important role in partial differential equation and harmonic analysis. Moreover, the properties of the solutions to integral system associated with HLS inequality have considerable theoretical significance and practical background. In this program, we will establish Liouville theorem for fractional Laplacian equations with the gradient terms and weighted function in Euclidean space and the upper space by using the method of moving planes or moving spheres. As applications, we shall investigate a prior estimate and the existence of solutions to the systems on some bounded domain. Then, we establish the weighted and reversed HLS inequalities in Euclidean space and the upper space. As applications, we shall investigate the asymptotic analysis of solutions to the weighted system associated with the extremal functions for above mentioned inequalities. Our results will improve the theories in partial differential equation and physics, and will provide the theoretical foundations to others related branch of learning. Therefore, this research project has both important scientific significance and research value.

分数阶Laplace算子是非局部算子的一个特例，近年来一直是偏微分方程和非线性分析领域关注度最高的研究热点之一。Hardy-Littlewood- Sobolev(HLS)不等式是分析学中一个基本不等式，在偏微分方程、调和分析等领域具有重要的应用价值。此外，HLS 不等式的极值函数相对应的积分方程组解的性质的研究具有很强的理论意义和实际背景。本项目首先利用移动平面法或移动球面法建立全空间和上半空间带梯度项的加权的分数阶Laplace方程组的Liouville 型定理。作为应用，讨论有界区域上相应方程组解的先验估计和存在性。然后，建立全空间上和上半空间加权的逆向HLS不等式。作为应用，讨论加权的HLS不等式的极值函数满足的积分方程组正解的渐进分析。本项目有助于丰富偏微分方程和物理学理论等，也将为其他学科相关问题的研究提供理论依据。

近年来，分数阶 Laplace 算子一直是偏微分方程和非线性分析领域的最热门的、关注度最高的研究热点之一。此外， HLS 不等式是利用偏微分方程理论研究曲率问题时必不可少的工具。本项目主要研究了分数阶 Laplace 椭圆方程（组）正解的对称性、单调性、存在性及分类问题；并且研究了 HLS 不等式的极值函数所对应的加权的积分方程组正解的渐进行为。本项目研究的具体内容为：. 1. 研究了上半空间带有非线性边值条件的一类非线性椭圆方程正解的分类问题。在超临界指数的条件下，证明了方程的所有正解都只依赖最后一个变量；当在临界指数的条件下，得到了正解的显式表达式。. 2. 讨论了上半空间含多调和延拓算子的积分方程组正解的分类问题，采用积分形式的移动球面法，建立了该积分方程组正解的 Liouville 型定理。. 3. 研究上半空间一类加权的积分方程组正解的最优的可积性，并分别在坐标原点和无穷远点建立了正解的渐进估计。. 4. 研究了 Heisenberg 群上加权的Hardy不等式、Rellich 不等式及其最佳常数问题。