The Presence Of Nonlinear Elliptic Equations And Qualitative Analysis

In this dissertation, we study the nonlinear elliptic equation's existence and qualitative properties:the existence of multiple solutions, radial properties of solutions, the priori estimates of solutions, asymptotic behavior of solutions and similar Kazdan-Warner identity. The main achievements contained here are as follows:1. Minimax methods and Morse theory are applied to study the biharmonic equation with asymptotically linear nonlinearities The existence of three nontrivial solutions is obtained.2. Minirnax methods are applied to study the p-Laplacian equation with mixed nonlinearities The existence of one nontrivial solution is obtained.3. Symmetry mountain pass theorem and topology are applied to study the fourth-order elliptic equations with perturbed nonlinearities The existence of arbitrary finite pairs nontrivial solutions is obtained.4. McOwen's methods tire applied to study the mean field equation with positive or negative total curvature The existence and asymptotic behavior at infinity of one nontrivial weak are obtained.5. The priori estimates, asymptotic behavior and classification of solutions for a fourth order elliptic equation with prescribing Q curvature are studied, by using integrand inequality, standard elliptic estimates and moving planes.6. The priori estimates, asymptotic behavior and convergence of solutions for a generalized Toda system arc studied, by using integrand inequality, standard elliptic estimates and blow-up analysis.This dissertation end with summary and research prospects.The innovations and features of this dissertation are as follows:(1) In research objection, we discuss the nonlinear elliptic equation or sys-tems of nonlinear elliptic which have physical and geometry background.(2) In the method, we successfully use the minimax method, Morse theory method and moving plane methods.(3) In content, we discuss the existence of nontrivial solutions for sonic classes nonlinear elliptic equations and analyze the qualitative properties of so-lutions for a class particular elliptic equation and elliptic systems.