In this paper, we focus on the existence results of radial solutions for a Dirichlet boundary value problem of the following augmented k-Hessian equation: \(\begin{aligned} \left\{ \begin{array}{ll} S_{k } (D^{2} u + \alpha I ) = \lambda b(\vert x \vert ) f(-u ) , & \quad i n \ \Omega , \\ u = 0 , & \quad on \ \partial \Omega , \end{array} \right. \end{aligned}\) where \( \Omega \) is an open unit ball in \( \mathbb {R}^{N},\) \( S_{k } (D^{2} u + \alpha I ) \) is an augmented k-Hessian operator, k is an integer, \( 1 \le k \le N < 2 k,\) \( \alpha \) is a constant \(\left( \alpha \ne 0 \right) \) , I is the unit matrix, \( \lambda \) is a positive parameter, b and f are continuous functions. The augmented term \(\alpha I\) in this paper can be positive or negative. It follows from the Guo–Krasnosel’skii fixed point theorem that there is at least one radial solution to the Dirichlet problem of the augmented k-Hessian equation for certain \(\lambda >0\) .
