几类非局部临界椭圆问题和相关变分法的研究

In recent years the study of nonlocal equations of elliptic type has attracted substantial attention, both for pure mathematical reasons and due to specific real-world applications. This family of equations appears increasingly in often diverse and novel contexts and applications. The objective of this project is to establish existence of ground state and multiple solutions to Nonlocal Critical Problems in Nonlinear Partial Differential Equations (in short, PDEs). The main idea is to extend known results for Critical PDEs and systems (involving the usual Laplacian) to nonlocal cases (involving the fractional Laplacian) and establish some new results for Nonlocal Critical Problems directly. It is planned to work on three kinds of problems: (1) Nonlocal critical equations, including the concave case, the linear case and the convex case for there is singularity or not on the boundary of the domain, respectively; (2) Nonlocal double critical problems, which is motivated by Li-Lin's open problem, is characterized by a singularity on the boundary of the domain; and (3) Nonlocal critical systems, again the study is divided into whether there is a singularity on the boundary of the domain or not. The research methodology consists in adapting the variational and topological techniques used in the “classical” critical case to the nonlocal fractional setting, using recently developed techniques, which were not available just a few years ago. Among the tools involved, we mention (nonlocal) Concentration -Compactness Method, Minimax Priciple and Index Theory etc.

近年来，椭圆型非局部问题受到了广泛关注，这不仅是由于数学上的意义，也是因为在现实中的具体应用，这类方程有着丰富的内容和大量的应用。本项目拟建立关于非局部临界椭圆问题的基态解和多解的存在性结果。主要目标是将临界方程和方程组(关于通常 Laplace算子)的一些已知结果推广到非局部情形(关于分数阶Laplace算子)，同时也直接建立一些关于非局部临界椭圆问题的新结果。具体解决三类问题：(1)非局部临界方程，在边界带有奇性或没有奇性的条件下，分别考虑凹项情形、线性情形和凸项情形；(2)非局部双临界问题，该问题受Li-Lin公开问题的启发，考虑边界带有奇性的情况；(3)非局部临界方程组，同样考虑边界有没有奇性两种情况。本项目的研究方案是利用最近发展的技术与方法，推广用于解决“经典”临界问题的变分与拓扑方法，使之适用于非局部临界问题的研究。(非局部)集中紧方法，极大极小原理和指标理论等将会被涉及。

近年来，椭圆型非局部问题受到了广泛关注，这不仅是由于数学上的意义，也是因为在现实中的具体应用，这类方程有着丰富的内容和大量的应用。本项目拟建立关于非局部临界椭圆问题的基态解和多解的存在性结果。主要目标是将临界方程和方程组(关于通常Laplace算子)的一些已知结果推广到非局部情形(关于分数阶Laplace算子)，同时也直接建立一些关于非局部临界椭圆问题的新结果。主要研究涉及分数阶Sobolev临界指数的方程组和双变量分数阶Sobolev不等式等，在适当的条件下，我们获得分数阶临界方程组基态解的存在性和类似惟一性结果，及其分数阶不等式的最佳常数的可达性等。本项目的研究方案是利用最近发展的技术与方法，发展用于解决“经典”临界问题的变分与拓扑方法，使之适用于非局部临界问题的研究。(非局部)集中紧方法，极大极小原理等将会被涉及。