Liouville-type Theorems And Isolated Singularities For Some Nonlinear Elliptic Equations

The system of coupled nonlinear Schr(?)dinger equations appears in many physical problems such as Bose-Einstein condensation and nonlinear optics.In recent decades,this system has aroused the interest of many famous mathematicians,and a large number of important research results have been obtainedThe first main content of this thesis is to study the qualitative properties of solitary wave solutions for coupled nonlinear Schr(?)dinger system and explore the impact of the coupling constants on these qualitative properties via elliptic equations theories.This is very important to understand the global dynamics of coupled nonlinear Schr(?)dinger system.More precisely,we study the Liouville-type theorems and a priori L? estimates of solitary wave solutions for coupled nonlinear Schr(?)dinger system,and connect these qualitative properties of solitary wave solutions with their stability and Morse indices The solitary wave solutions we consider are possibly sign-changing,and our results can be applied to the subcritical,critical and supercritical cases of the system.This seems to be the first result for a priori estimates of sign-changing solitary wave solutions to Schr(?)dinger system.We also study the Liouville-type theorems for positive solutions of a class of quasilinear elliptic systems.In particular,our results can be applied to non-cooperative quasilinear Schr(?)dinger-type systemThe second main content of this thesis is the isolated singularities of nonlinear elliptic equations.This kind of problem is of great significance not only in the theory of PDEs but also in the construction of constant curvature metrics with isolated singularities in conformal geometry.Firstly,we study the classification of isolated singularities of positive solutions to the fractional Lane-Emden equation and give the exact asymptotic behavior of positive solutions near isolated singularities.We also give a classification of isolated singularities of positive solutions and the exact asymptotic behavior of positive singular solutions to a biharmonic Lane-Emden equation.In the classical Laplacian case,the similar results are obtained by Caffarelli,Gidas and Spruck in their celebrated papers(Comm.Pure appl.Math.,1981,1989).However,compared with results of Caffarelli,Gidas and Spruck,the powerful tool of ODEs analysis is a missing ingredient for fractional equation and the biharmonic equation has no the maximum principle,which pose great difficulties to the study of isolated singularities of the corresponding equation.In this thesis,we present a new method based on the monotonicity formula to completely solve the isolated singularities of these two kinds of equations in the case we are consideringFinally,we study the isolated singularities for coupled nonlinear Schr(?)dinger sys-tem.In the subcritical case,we characterize the singularities of positive solutions of such system and prove the nonexistence of semi-singular positive solutions.We also obtain the asymptotic behavior of both two components of positive solutions near isolated singularities.