Qualitative Research Of Hénon And Hardy Elliptic Boundary Value Problem

Nonlinear elliptic differential equations and systems play an important role in industrial production and scientific processes.In many fields,it is necessary to establish a suitable mathematical model,which is described by differential equation,to ensure the safety and efficiency of engineering and project.Variational method is one of the effective methods to study this kind of problems.We can solve the problems of nonlinear elliptic boundary value by studying the functional extremum problems.In this paper,we study the existence and multiplicity of G-symmetric solutions of Hénon type and Hardy equation(system of equation)in G-symmetric functional spaces.The paper is divided into six chapters as follows:In Chapter 1,we introduce the research background and research status of nonlinear elliptic equations,put forward some problems with certain research value,and which need to be solved,and then we introduce the main contents of the whole paper briefly.In Chapter 2,we explain the basic theorem and formula symbols that need to be used in this paper.In Chapter 3,the symmetry solutions of a semilinear elliptic equation with Hénon critical exponent exponent and perturbation exponent is investigated by the mountain pass theorem,symmetrical critical principle and concentration compactness principle.The existence and multiplicity of symmetric solutions of the problem in G-symmetric space are studied under the condition of establishing appropriate limit growth for perturbation term.In Chapter 4,on the basis of the third chapter,the research is further extended to the system,and the symmetry solutions of a class of semilinear elliptic systems with Hénon critical exponent are investigated.After obtaining the conclusion of the solution of the equation,we use the similar method to qualitatively study the system of equations in the G-symmetric space.In Chapter 5,we summarize the experience and conclusions of the third chapter and the fourth chapter,and investigate the symmetric solutions of a class of singular quasilinear elliptic systems with homogeneous nonlinear critical exponents.The existence and multiplicity of G-symmetric solutions of the Hardy-type ellipticboundary value problems under different conditions are obtained.In Chapter 6,the main content of the paper is summarized,and we put forwards some prospects for the existing problems and the future work arrangements.On the one hand,nonlinear differential equations have great academic value,and promote the theoretical research of this kind of problems.On the other hand,on the basis of the qualitative research results of the mathematical model,it can be combined with the concrete project,and has certain application prospect.