Systems Of Elliptic Equations Involving Multiple Inverse-square Potentials And Critical Exponents

In this paper，a system of elliptic equations is investigated，which involves multipleinverse-square potentials and critical Sobolev exponents．The problem itself and thebackground and meaning of this elliptic systems is presented in the introduction，and wealso introduce some related preliminaries and definitions．Owing to the systems of elliptic equations involving multiple inverse-squarepotentials and critical exponents， the corresponding energy function lose thecompactness and it lost Palais-Smale condition．Namely，the Palais-Smale sequenceconvergent only for some valves of c．So firstly we need to establish local Palais-Smalecondition before proving the existence of positive solutions to the systems．Therefore，in chapter2， The best Sobolev constant related to system is studied，which is verifiedthat depends closely on λ and μ and to be independent of the location of singularpoints a and b．Then we bulid the corresponding functionals of the local Palais-Smalecondition of the system which mainly use the concentration-compactness principle．In chapter3， we mainly prove the existence of its positive solutions(Mountain-pass solution)．Before this，the relationship between the relevant bestconstants of this elliptic systems and the estimated style of extremal functions aregiven．Then the existence of positive solutions to the systems of elliptic equations isproved by using these results and the mountain-pass argument．At last，we prove that the systems of elliptic equations has a pair of sign-changingssolutions on the basis of the mountain-pass-type positive solution．Some of thefunctional theoretical knowledge and a few inequalities are used in this part．...