On The Critical Phenomena And Stabilization Of A Kind Of Nonlinear Systems

This paper investigate the development trend and state of a kind of nonlinear systems originated from nature and society. The problem of nonlinearities is an important research object faced by the field of nature, engineering technology and the economics.This paper deals with the boundary problems of nonlinear partial differential equations with singular critical exponent abstracted from the problems with essential backgrounds of physics, biology and sociology. By using critical point theory and variational methods, we will discuss the existence of the critical state and its reliability on the parameters for the critical state of the nonlinear systems, and we will also discuss the stability and the multiplicity of the critical state and investigate the effect of environmental parameters on the systems to find the regulations of the systems.In these days, the most important and difficult problems in researching nonlinear systems are the difficulties of losing compactness in the mathematical model, and the elliptic equations with critical exponents term are of some importance in the nonlinear partial differential equations originated from nonlinear system problems. For losing the compactness, the corresponding functional will not satisfy (P.S) condition, and thus there are many difficulties in application of the standard variational methods. This paper will discuss the existence and property and the the effect of the domain on the number of the solutions for the nonlinear partial differential equations with singular critical exponent by using concentration-compactness principle and linking methods and accurate estimation of the energy of the functional.The main results and novelties of the paper are as follows:Firstly, we study the existence of nontrivial solution for the following nonlinear equations with singular pontential By using variational methods and critical point theory, we construct linking type critical value of this kind of singular elliptic equations, with the properties of the eigenvalue, we estimate the critical value and prove the local (P.S) condition holds to show the existence of the nontrivial solution.Secondly, we consider the existence of infinitely many solutions for a class of quasilinear equations with Hardy critical exponent term with the aids of variational methods and concentration-compactness principle, we overcome the losing of compactness for embedding mapping and use the perturbation methods to construct the existence of infinitely many critical value, thus we obtain of the the existence of infinitely many solutions.Thirdly, we will concentrate on the multiplicity of the positive and minimal sign-changing solutions of the following quasilinear equations with Hardy-Sobolev critical exponent in bounded domain By Lusternik-Schnirelmann category theory, we showthe relation the number of positive solutions with the topology of the domain. Moreover, With assumption of symmetry on the domain, we show the effect of the domain topology on the number of minimal nodal solutions and give a characteristic of the number of solutions.Finally, we consider the multiplicity of the positive and minimal sign-changing solutions of the critical singular equation involving the Caffarelli-Kohn-Nirenberg inequalities of the type With the similar method used above, we obtain the number and properties of the solutions.By using accurate analysis technique and calculation, this paper shows the existence and multiplicity of positive, sign-changing solutions of the corresponding equations for nonlinear system with critical exponent and singular terms whose variational functional loses compactness. The results obtained here improve the results and generalize the methods and techniques by former mathematicians and the results help to character the solution set for the nonlinear system and provid the theory for researching the stability problems of nonlinear systems.