This paper mainly studies the existence, multiplicity and asymptotic behavior of k-convex radial solutions for the following Dirichlet boundary value problem of the augmented k-Hessian equations: \(\begin{aligned} \left\{ \begin{array}{l} S_{k} \left( D^{2} u + \alpha I \right) = \mu p\left( |x| \right) g \left( -u \right) , \quad i n \ \Omega , \\ u = 0 , \quad on \ \partial \Omega , \end{array} \right. \end{aligned}\) where \( \Omega \) is an open unit ball in \( {\mathbb {R}}^{N} \) , \( S_{k} \left( D^{2} u + \alpha I \right) \) is the augmented k-Hessian operator, \( \frac{N}{2} < k \le N \) , \( D^{2} u \) is the Hessian matrix of u, \( \alpha \) is a constant \(\left( \alpha \ne 0 \right) \) , I is the unit matrix, \( \mu \) is a positive parameter, and p and g are continuous functions. The augmented term \( \alpha I\) in this paper can be positive or negative. Our analyses are based on the Krasnoselskii fixed point theorem.
