Existence And Multiplicity Of Positive Solutions To Elliptic Systems Involving Critical Exponents

In this paper, we deal with the following class of systems of nonlinear singular elliptic equations, involving Caffarelli - Kohn - Nirenberg inequality and coupled by a nonlinear term (?)where ? ? R N ( N≥3) is a bounded domain with smooth boundary (?) is the critical Sobolev - Hardy exponent, (?)is the best Hardy constant, (?). We are interested in the existence and multiplicity of the positive solutions to the elliptic systems.Firstly, we give a brief introduction about the problem, research background, and some previous research results. Then we give some notations and definitions which are closely related with this paper. Secondly, since the lack of compactness of the embedding(?), we cannot use the standard variational argument directly, so we have much difficulty in dealing with this problem by variational method. To overcome this difficulty, we established local Palais-Smale conditions for the corresponding energy functional of the systems. Meanwhile, because of the specificity and complexity of the elliptic systems involving multiple critical exponents, we need to study the relationship between the best constant. Concentration-compactness principle and variational inequalities are fully used in this part. Furthermore, we make full use of the cut-off techniques to the corresponding energy functional of the problem. Then, as the energy functional J is not bounded below on E: = H 01 (?)×H01(?), it is useful to consider the functional on the Nehari manifold, we give some important properties about Nehari manifold.Finally, based on the results above, the existence and multiplicity results of positive solutions (Mountain-pass solution) to elliptic equations (1.1.1) are obtained by variational method and analytical technique under certain conditions.