Existence of k-Convex Solutions for the k-Hessian Equation

This paper considers the following Dirichlet boundary value problem of the k -Hessian equation: {S-k (sigma (D(2)z)) = lambda b(|x|)?(-z), in omega, z = 0, on & part;omega,where lambda > 0, omega stands for the open unit ball in R-N, 1 <= k <= N is an integer, and S-k (sigma (D(2)z)) is the k -Hessian operator of z. We obtain the existence results of k -convex radial solutions of the k -Hessian problem for lambda belonging to an open interval. Our main approach is the Guo- Krasnosel'skii fixed point theorem in a cone.