Polyharmonic $k-$Hessian equations in $\mathbb{R}^N$

Abstract

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.