Research On Radially Symmetric Solutions Of K-Hessian Type Equations (groups

As the theory of nonlinear partial differential equations matures,geometric analysis has developed rapidly.The k-Hessian type equations are derived from some geometric and physical problems,which not only have important research value in the theory of nonlinear partial differential equations,but also play a huge role in practical applications such as optimal transportation problems,geometric optics,computer graphics and medical imaging.It is also of great significance to study a series of properties of solutions to k-Hessian type equations or systems,such as existence and multiplicity.In this thesis,we study some properties of radial solutions of k-Hessian equations or systems,which are divided into two parts.In the first part,the existence and multiplicity of radial convex solutions to a class of fully nonlinear system weakly coupled by ki-Hessian equations are studied via a fixed-point theorem,and then some topological degree methods are applied to study the uniqueness,nonexistence and related eigenvalue problem of radial solutions to a power-type ki-Hessian system.In the second part,a necessary and sufficient condition for the existence of entire kadmissible subsolutions to a kind of k-Hessian type equations with gradient terms is studied,which extends the classical Keller-Osserman condition to the k-Hessian operator with a gradient term and can also be seen as a necessary and sufficient condition for the existence of radial solutions in the whole space.