On The Singular Elliptic Systems Involving Multiple Critical Sobolev Exponents

In this paper, a class of singular elliptic system is investigated, which involves multiple critical Sobolev exponents and Hardy--type terms. Firstly, we study the problem itself and the background of an overview and introduce some related preliminaries. This problem appeared in variational problems of mathematical physics equations in the late seventeenth century at first. In the eighteenth century, by Euler, Lagrange and others'working it gradually formed a branch of mathematics - variational method to resolve mathematical physics problems. It is also the main method to solve such problem in this paper.Secondly, we give the corresponding variational functionals of the local Palais-Smale condition and the relationship between the relevant best constants of the problem, then prove the existence of its positive Solutions (Mountain-pass solution) by using these results. The main difficulty is that it lost Palais-Smale condition due to the problem contains a number of critical exponents. Therefore, we need to establish local Palais-Smale condition. Since the particularity and complexity of the equations, we have to identify the relationships between the best constants related to them. We mainly use the concentration-compactness principle and variational inequality and analytic techniques to the corresponding energy functional of the problem. Here we need to take attention to the critical items particularly.Then we study the asymptotic properties of its non-trivial solutions at the singularity points. This is one of the important previous foundations of the equations that we study the existence of sign-changing solutions. Because of the singularities of the problem, we have to overcome the singularities of its non-trivial solutions at the singular points. We use the Moser iteration method to prove the asymptotic properties of its non-trivial solutions at the origin (ie, singularity) in this paper.Finally, we study the existence of sign-changing solutions of the equations. Based on the variational structure of this problem, the existence of sign-changing solutions to the system is established by the same variational methods. Some of the functional theoretical knowledge and a few inequalities are used in this part.