  This work is devoted to the study of radial solutions to the elliptic problem
\begin{equation}\nonumber \Delta^2 u = (-1)^k S_k[u] + \lambda f, \qquad x \in
B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet
boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x
\in \partial B_1(0), \end{eqnarray} or Navier boundary conditions
\begin{equation}\nonumber u = \Delta u = 0, \qquad x \in \partial B_1(0),
\end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$
elementary symmetric polynomial of eigenvalues of the Hessian matrix and the
datum $f \in L^1(B_1(0))$. We also study the existence of entire solutions to
this partial differential equation in the case in which they are assumed to
decay to zero at infinity and under analogous conditions of summability on the
datum. Our results illustrate how, for $k=2$, the dimension $N=4$ plays the
role of critical dimension separating two different phenomenologies below and
above it.
