Liouville-type Theorems And Properties Of Solutions For Some Degenerate Elliptic Equations(Systems)

The classical Liouville theorem is pointed that bounded harmonic functions defined in the whole space must be a constant.In recent decades,Liouville theorem has been widely studied and extended to various equations(systems)by domestic and foreign scholars.At the same time,this theorem has been extended by domestic and foreign scholars to study the existence and nonexistence of solutions to various equations(systems)(Liouville-type theorem),symmetry and monotonicity of solutions,etc.In addition,the sub-elliptic operator ?H on the Heisenberg group is degenerate on everypoint and has extensive applications in geometric control,non-holonomic mechanics,financial mathematics,medical imaging and theoretical physics.This paper is mainly based on truncation function technique,prior estimates,energy methods,fixed point theory and nonlinear functional analysis theory,etc.,the existence and nonexistence of solutions of several sub-elliptic equations(systems)on Heisenberg group are studied,and the uniqueness and symmetry results of solutions of a class of sub-elliptic equations are proved.The main specific work is as follows:In Chapter 2,we consider the systems of sub-elliptic inequalities on the Heisenberg group(?)and(?)where ?H denotes the Heisenberg Laplacian in the Heisenberg group Hn(n?1),and hi(i=1,2,3)are some non-negative functions,and ?(?)Hn is an unbounded domain.By using the energy method,the truncation function and some analysis techniques,we prove that,under suitable conditions on hi,p,q,s and ?,the above sub-elliptic systems do not possess positive solutions(Liouville-type theorem)in the whole space or half space.In Chapter 3,we consider the following sub-elliptic equation with singular nonlinear terms:(?)where ?(?)Hn is a smooth bounded domain,?>0 and h? 0.We first use the Schauder's fixed point theorem and approximating method to prove the existence of solutions to the above equa-tion.We then obtain the uniqueness result by proving a weak comparison principle and further deduce that the solution is cylindrically symmetric under some necessary structural conditions on ? and h.In Chapter 4,we are concerned with the following Schrodinger-Poisson type sub-elliptic system with the critical exponent on the Heisenberg group H1:(?)where ?H is the Kohn-Laplacian on the first Heisenberg group H1 and ?(?)H1 is a smooth bounded domain,1<q<2,??R and ?>0 some real parameters.By the Green's represen-tation formula,the concentration compactness and the critical point theory,we prove that the above sub-elliptic system has at least two positive solutions for ?<S × meas(?)-1/2 and ? small enough,where S is the best Sobolev constant.Moreover,we show also that there is a positive ground state solution for the above system.