A Sort Of Full Nonlinear Asymptotic Dirichlet Problems On Cartan-Hadamard Manifolds

The existence and uniqueness of global solution to a sort of fully nonlinear partial defferential elliptic equations, which are posed on complete non-compact manifolds, are central topics of this thesis. It is concept of viscosity solution which was defined in M. G. Crandall and P. L. Lions [9] that makes the great progress of theory in fully nonlinear equations. At present, the methods of solving fully nonlinear equations can roughly fall into two categories.The first one is Perron method. This method is mainly characterized by the usage of comparison theorem of sub-and super-solutions. Following this way, M. G..Crandall, P.L.Lions [9], H.Ishii [22] have solved Hamilton-Jacobi equations, H.Ishii [21], [23]have solved elliptic equations,M.G.Crandall, P.L.Lions [10] and H.Ishii [20] have solved parabolic equations.The second one is continuity method. Initial works are given by Luis A. Caffarelli, L.C.Evans and Xavier Cabre. The tough goal in this method is to get the prior estimate of solution. When the prior estimate of solution is in hand, one can make use of Leray-Schauder fixed point theorem to get existence. We can see this philosophy in Xavier Cabre [14] [13], L.C.Evans [15] and Luis A. Caffarelli [16] [17].The simplest elliptic equations on complete non-compact manifolds is the Laplace equation, which were initially studied by Hyeong.In.Choi [6]. He got the existence of global solution of the boundary problem by assuming a type of convex condition on the manifolds. This question was eventually settled by the works of M.T.Anderson [1] and D.Sullivan [25].Soon after this works, M.T.Anderson and Richard.Schoen [2] construct barrier functions directly to solve this problem.Recently, Daniel Azagra, Juan Ferrera, Beatriz Sanz [4] and Shige Peng, Detang Zhou [24] have proved comparison theorem on compact domain of manifold under a coercive condition. However, our equations do not cater for this condition, since the Pucci or Laplace-like equations, which in most situation, do not have zero-order term. We solve this type equations by introduce a auxiliary function, and get the comparison theorem, and then solve a large sort of equations. This thesis is organized in the following way:In chapter 1, a brief reviews of fully nonlinear equations on Rn and an asymp-totic Dirichlet problem concerning Laplacian equation on Cartan-Hadamard manifolds have been given. And it also includes some recent progress of viscosity solutions on Riemannian manifolds.Chapter 2 introduces the main fully nonlinear equations and circustance, which we are going to solve. It should be emphasized here is that non-coercive monotonic conditon and growth condition.In chapter 3, a auxiliary function on pinched Cartan-Hadamard manifolds has been intensively studied. This function is of great importance, because it has global propeties of convex, Laplacian bounded from below by a positive constant and with bounded gradient. This properties can not be possessed simultaneously by a single function defined on Euclidean space.In chapter 4, we present our main theorem. Firstly, we construct comparison theorem for sub-solution and super-solution of asymptotic Dirichlet problem. Then, by using this key tool, we employ the perron method to get the existence and uniqueness of the equation.In chapter 5, we mainly study an important fully nonlinear equation, Pucci equa-tion. By searching sub-solution and super-solution, we got the unique solution of such asymptotic problem, and, so, a bounded non-trivial solution on the complete noncom-pact Riemannian manifolds.