Nontrivial Solutions To Elliptic Systems Involving Multiple Critical Exponents

In this paper, we study the following system: (?)whereΩ? N ( N≥3) is a bounded domain with smooth boundary ?Ωsuch that(?) is a critical exponent andμ:= ( N2?2)2 is the best Hardy constant. This is a singular elliptic system involving multiple critical exponents. Partial Differential Equation (P.D.E for short) plays an important role, whether in Mathematical Subject itself, or in the other fields of real life. From the perspective of Mathematical Subject itself, P.D.E is an important branch of mathematics. The finding of solutions of partial differential equations prompts the development of other parts of mathematics, such as Function Theory, Variational Method, Ordinary Differential Equation, Series Spread and so on. From this part, P.D.E is the central part of mathematics. But in the real life, the study of Natural Science and Engineering Technology, which view Mechanics, Biology, Chemistry and Physics as the research background, can be changed to the study of P.D.E. For example, the movement process of mechanics is a typical partial differential equation. As the development of Physical Science, the application of P.D.E is wide. Therefore, the study of P.D.E is not only the need of Mathematical Subject, but also the need of real life. Elliptic systems are the important part of P.D.E. Elliptic systems can be applied in Fluid Dynamics, Elastic Dynamics, Electromagnetism, Geometry and Variational Method. For Elliptic system, we pay attention to the existence of positive solutions, multiple solutions, or nontrivial solutions, and search the properties of some solutions. It's easy to study the solutions of linear elliptic system, but in the application, we more emphasize the solutions which satisfy certain conditions. We use Finite Difference Method, Variational Method, Integral Equation Method and so on to study the Elliptic systems. In this paper,we want to prove the existence of nontrivial solutions of the system for large ranges of ai (1≤i≤3). By the standard elliptic argument we know that nonzero critical points of functional J (u , v )correspond to nontrivial solutions of the system.Firstly, we introduce the background of the question and some knowledge. This is a variational problem, and we use Variational Method. We get some properties of the operator L and attain the first eigenvalue of L by using Hardy inequality. From Young inequality, we can define some best constants. In this section, the key is to find the function which achieves the best constant.Secondly, the classic Variational Method can not be used because of the energy functional. So, by the standard elliptic theory and Brezis-Lieb Lemma, we get the local (PS) condition and by direct calculation, we get some asymptotic properties of the extremal function. From these results, we get the nontrivial solutions of the system.At last, by using variational inequalities, we analysis the energy functional, and using Linking Theory, we get the proof of Theorem 1.3.1, Theorem 1.3.2 and Theorem 1.3.3. By Variational Method and Critical Theory, we prove that the system has vk pairs of nontrivial solutions, where vk denotes the multiplicity ofλk .