  This work is focused on the study of the nonlinear elliptic higher order
equation \begin{equation}\nonumber \left( -\Delta \right)^m u = S_k[-u] +
\lambda f, \qquad x \in \mathbb{R}^N, \end{equation} where the $k-$Hessian
$S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of
eigenvalues of the Hessian matrix of the solution and the datum $f$ belongs to
a suitable functional space. This problem is posed in $\mathbb{R}^N$ and we
prove the existence of at least one solution by means of topological fixed
point methods for suitable values of $m \in \mathbb{N}$. Questions related to
the regularity of the solutions and extensions of these results to the nonlocal
setting are also addressed. On the way to construct these proofs, some
technical results such as a fixed point theorem and a refinement of the
critical Sobolev embedding, which could be of independent interest, are
introduced.
