Behaviors And Critical Surfaces For The Solutions To Elliptic Systems Involving Different Hardy–type Terms

In this paper,a system of semilinear singular critical equations and a corresponding system of quasilinear singular critical equations are investigated,which is divided into three chapters.In the first chapter,we introduce the problems itself and their backgrounds.Moreover,we introduce the used symbols,definitions,and main results of this paper.Finally,we explain the structure of this paper.In the second chapter,a system of semi-linear singular critical equations is studied,which involves strongly-coupled terms and different Hardy terms.Firstly,by constructing transition variables,cut-off functions and combining inequality scaling methods,we tansform the related integral form of the solutions of weighted functions,and then we obtain an asymptotic estimate at origin of weak solutions by the Moser’s iteration method and variational methods.Furthermore,optimal estimates on asymptotic behaviors of ₀v in a weak solution（u₀,v₀）are obtained to the problem（1.1.1）,especially forβ<2,the asymptotic behaviors of weak solutions are proved above and below the critical surface respectively,and then it is found that the singularities of ₀v are different above or below the critical surface.In the third chapter,a system of quasilinear singular critical equations is studied,which involves strongly-coupled terms and different Hardy terms.Firstly,preliminary asymptotic estimates of weak solutions to equations（1.1.2）are established.Secondly,a more general equations than system（1.1.2）is given,and then we establish comparison principles for weak subsolutions of the system both on bounded domains and exterior domains.Next,we make an optimal asymptotic estimate for the positive weak solutions to equations（1.1.2）.Finally,from the results obtained,the optimal asymptotic estimates on the gradient of positive weak solutions to equations（1.1.2）can be obtained.Another critical surface is also found,above and below which the asymptotic properties at the infinity of positive weak solutions are different.