著名的 Caffarelli-Kohn-Nirenberg(CKN) 不等式(Compos. Math., 1984) 包含了经典的 Sobolev 不等式和 Hardy 不等式作为特例，它在泛函分析、偏微分方程等数学分支研究里是一个非常重要的不等式。 CKN 不等式中的等号是否取到、最佳常数为多少、达到函数是什么样子的或者具有什么样的性质等问题是近三十多年来分析与非线性方程领域中许多专家非常关心的问题，很多著名的数学家在这方面做出了大量杰出的贡献。本文旨在利用变分法和椭圆方程的理论，研究与CKN不等式有关的含 Hardy-Sobolev 临界指数的方程和方程组。包括最小能量解的存在性和非存在性问题，正解的存在性问题，无穷多解、变号解的存在性问题，以及解的正则性、对称性、衰减估计等性质的研究。首先，我们考虑一类有界区域上涉及 Hardy-Sobolev 临界指数的非线性Schr\"odinger方程，研究了方程形式上满足``最高次幂方项的系数是负的"这种情形正解的存在性问题。在国际上给出了 Li Yanyan 和 Lin Changshou 在文献(Arch. Ration. Mech. Anal.,2012) 中提出的公开问题的第一个回答。另外对于带有双 Hardy-Sobolev 临界指数项的次临界扰动问题，我们研究了基态解或正解的存在性。建立了对一般区域均适用的一系列重要的插值不等式，并成功应用来证明了锥上的一类 CKN 不等式的最佳常数是可达的。同时将上面问题的研究成果推广到无界区域的情形，这是这类方程在无界区域（非极限区域）上的首次尝试。同时在 $\R^N$ 上考虑了有多重 Hardy-Sobolev 临界指数的方程，发展了 Lions 的集中紧思想，并结合扰动方法研究了基态解的存在性问题，系统地研究了正解的正则性、对称性、衰减估计等性质。另外，我们还研究了椭圆系统的情形，这是对涉及 Hardy-Sobolev 临界指数的椭圆系统方面的第一次尝试。我们首次获得了这类系统基态解的存在性、唯一性、对称性、正则性、衰减性估计等一系列成果。其中的一些结果将成为研究这类系统的根本性定理。
The well known Caffarelli-Kohn-Nirenberg(CKN) inequality (Compos.Math., 1984) include the classical Sobolev inequality and Hardy inequality as its special cases. It plays an crucial role in much research such as functional analysis and partial differential equations.Whether the equal sign in CKN inequality can hold? What is the value of the best constant? If this sharp constant can be achieved, which is the extremal function or possesses what kind of properties? These problems are very concerned in the last thirty years and many famous mathematicians have made a large number of contributions in this related hot spot.This thesis aims to use the variational method and the elliptic theory to investigate the equations and systems involving Hardy-Sobolev critical exponents, which are close relate to the CKN inequality. We will study the existence or nonexistence of the least energy solution, the existence of positive solutions, the existence of infinitely many solutions or sign-changing solutions, the regularity, symmetry and decay estimation about the solutions, etc.Firstly, we consider a family of nonlinear Schr\"{o}dinger equations involve Hardy-Sobolev critical exponent in a bounded domain.We will study the existence of positive solution to those equations whose form satisfies `` the coefficient of the highest power term is negative". These provide some partial answers, but as far as we know the first ones to an open problem proposed by Li Yanyan and Lin Changshou in the remarkable paper (Arch. Ration. Mech. Anal.,2012). We also consider a perturbation nonlinear elliptic PDE involving two Hardy-Sobolev critical exponents, we study the existence of ground state solution or positive solution. We will establish a sequence of interpolation inequalities which are applicable to the general domains. As an application, when the domain is a cone, we prove that the best constants of a family of CKN inequalities are achievable. And then we extend the results above to the unbounded domains, it is the first attempt on the unbounded domain (not a limit domain). We also consider a PDE in $\R^N$ involving multiple Hardy-Sobolev critical exponents. We study the existence of ground state solution by developing the ideas of concentration compactness principle by Lions and the perturbation method. Meanwhile, we study the regularity, symmetry as well as the decay estimation about the positive solutions.Besides, we also consider the elliptic system case and it is also the first try on this kind of system involving Hardy-Sobolev critical exponents. We obtain a sequence of results for the first time such as the existence, uniqueness, symmetry, regularity and decay estimation of the ground state solution. We note that some of these results will become fundamental.
