Study On The Boundary Value Problems Of P-q Type Elliptic Systems Involving Critical Exponents

Nonlinear elliptic differential equations and systems are widely used in physics,geometry,biology and other disciplines,and it plays an important role in industrial production and the scientific process.This paper investigates nontrivial solutions of p-q type elliptic equations and systems with critical exponential term.On the one hand,the p-q type elliptic boundary value problems with critical exponent has great academic value,which promotes the theoretical research of this kind of problems.On the other hand,it lays a solid theoretical foundation for the application of physics and chemical reaction control.The full text is divided into six chapters:The first chapter is the introduction part.The research background and research status of the p-q elliptic equations and systems are presented.It puts forward some unsolved and valuable problems.What's more,it briefly introduces the main contents.The second chapter explains the basic symbols which are used in this research,and gives the related definitions and preliminary lemmas.In the third chapter,it makes a research about a class of p-q elliptic equations with Hardy-Sobolev critical exponents,and it proves the existence of positive solutions of the equation by using variational methods and mountain pass lemmas.And combining with the index theory,it also obtains the infinite multiple solutions of the equations.In the fourth chapter,by introducing the G-symmetric function space,the author conducts an investigation about the G-symmetric solution of a class of p-q type elliptic equations with Hardy term and Sobolev critical exponents in symmetric function spaces.The appropriate restrictions on the parameters are given,and the cases with the perturbation term and the nondisturbance term are discussed.It proves the existence and multiplicity of the G-symmetric positive solutions.This chapter generalizes the G-symmetric solutions to the p-q type in the equation.The result is completely original and it has theoretical value.In the fifth chapter,the research studies the multiplicity of solutions for a class of p-q type elliptic systems with critical homogeneous nonlinearity.It adopts Nehari manifolds,Lusternik-Schnirelmann category theory and analytical techniques to get the multiple positive solutions and tries to combine the critical homogeneous nonlinearity with the p-q type elliptic systems to generalizes the existing conclusions.The sixth chapter summarizes the major findings and gives some limitations of this present studies and some suggestions for future work.