The Existence And Multiplicity For The Nonlinear Critical Equations And Systems

The existence and multiplicity for the nonlinear critical equations and systems havebeen widely applied to astrophysics, conformal geometry, nonlinear optics and quantummechanics. Therefore, they are the main topics in nonlinear analysis for the last decades.Many mathematicians have made great contributions in this area.In this thesis, we study the existence and multiplicity for some nonlinear criticalequations and systems via Lyapunov-Schmidt reduction method, elliptic theory and vari-ational method. These problems are:1. the multiplicity for the polyharmonic equationwith the critical exponent,2. the multiplicity for the polyharmonic Yamabe type equation,3. the multiplicity for the non-cooperative critical Schro¨dinger system,4. the existencefor the quasilinear critical Schro¨dinger system.Firstly, we study the multiplicity for the polyharmonic equation with the criticalexponent.Assume that the non-autonomous term K has a local positive maximum, we con-struct k positive non-radial bubbles whose centers are divergent to the infinity as k is largeenough in the Lyapunov-Schmidt reduction method. The existence of infinitely manypositive non-radial solutions for the polyharmonic non-autonomous equation is provedby gluing and perturbing these k bubbles.Secondly, we study the multiplicity for the polyharmonic Yamabe type equa-tion.We construct k non-radial negative bubbles by the Lyapunov-Schmidt reductionmethod where k is the control parameter. The centers of those bubbles are all concen-trated inside the unit circle as k grows to the infinity. The existence of infinitely manynon-radial sign-changing solutions for the polyharmonic Yamabe type equation is provedby gluing and perturbing the k negative non-radial bubbles and1positive radial bubble inaddition.Thirdly, we study the multiplicity for the non-cooperative critical Schro¨dingersystem.Given any negative coupling coefcient β <0, we set up a series of singular esti-mates in R3by the superlinear structure of the coupling terms. The existence of infinitely many non-radial positive solutions for the non-cooperative critical Schro¨dinger system isproved by combining the Lyapunov-Schmidt reduction method, the fixed point methodand the singular estimates.Finally, we study the existence for the quasilinear Schro¨dinger system with thecritical exponent.Combining the maximum principle in elliptic theory and the concentration-compactness principle in variational methods, we prove the existence of a non-trivialsolution for the quasilinear Schro¨dinger system with the exponent. By Schauder esti-mates and Moser iterations on this non-trivial solution, the existence of a positive groundstate solution is proved.