Solutions Of Nonlinear Elliptic Equations And Systems Involving Critical Hardy-sobolev Exponents

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 so-lutions or sign-changing solutions, the regularity, symmetry and decay estimation about the solutions, etc.Firstly, we consider a family of nonlinear Schrodinger 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 RN 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.