| As a completely nonlinear second-order partial differential equation,k-Hessian equation has a wide application in the fields of differential geometry,fluid mechanics and other applied disciplines.At the same time,the k-Hessian equation is reduced to the classical Laplace equation when k=1 and the classical Monge-Ampere equation when k=n.In this thesis,we consider the existence and asymptotic behavior of k-convex solutions for two classes of k-Hessian equations.In Chapter 1,by using the Karamata regular variational theory and the upper and lower solution method,we study the existence and asymptotic behavior of strictly k-convex solutions for a class of singular k-Hessian problem.Firstly,we obtain the existence and asymptotic behavior of strictly k-convex solutions for this problem when the nonlinear term f=s-γ.Secondly,the existence of strictly kconvex solutions for this problem is obtained when the nonlinear term f is a more general function.Finally,the existence and asymptotic behavior of strictly k-convex solutions for this problem is also obtained when the weight function b is singular at the boundary.In Chapter 2,by applying the fixed-point theorem,we study a class of coupled k-Hessian system with nonlinear operator.Firstly,we obtain the existence of radial k-convex solutions for this coupled system.In addition,the asymptotic behavior of radial k-convex solutions for this coupled system is also obtained,which dependent on parameters. |
