| In this paper,we study a class of quasilinear critical elliptic systems involving p-Laplacian operator,critical Sobolev nonlinearities and multiple singular potential terms.This can help us to describe the balance and stable state of magnetic field,force field and reaction-diffusion phenomena,study and master their properties,and apply them to engineering technology.By using the variational method,under certain conditions,we prove the existence and nonexistence of minimizers to the related best Sobolev constant,at the same time,we give the conditions for the existence of ground state solutions of the systems.In the first chapter,we introduce the elliptic systems which to be studied,briefly summarize its research background,and list some scientific research achievements which have great inspiration for the elliptic systems studied in this paper.we explain the symbols and mathematical definitions in this paper,and generalize the conclusions of this paper into theorems.Finally,we supplement the structure of this paper.In the second chapter,we consider that the elliptic systems studied contain the critical Sobolev nonlinearities and multiple singular potential terms,and the region studied is Ρ~N.We establish the local Palais-Smale condition(abbreviated as local(PS)_ccondition)of the systems by using the principle of concentrated compactness,and ensure that the associated critical sequence also has strong convergence.In the third chapter,we study the asymptotic properties of minimizers at singular points.In this chapter,we use the research results and methods in some papers for reference,and combine the problems studied in this paper,we get the required results,which is the preparation for the theorems proving in the fourth chapter.In the fourth chapter,we prove the theorems in this paper.This paper includes two theorems about the existence and nonexistence of minimizers to the related best Sobolev constant,and gives the conditions that the existence of ground state solutions should satisfy.Through the preparation of the contents of the first three chapters,as well as the lemmas in this chapter,combined with the knowledge of functional analysis and mathematical analysis,we prove that the theorems in this paper is tenable. |
