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

TL;DR

This paper investigates higher order nonlinear elliptic equations involving the $k$-Hessian operator in $ abla^2 u$, establishing existence of solutions using topological fixed point methods and exploring regularity and nonlocal extensions.

Contribution

It introduces new fixed point techniques and refined Sobolev embeddings to prove existence of solutions for polyharmonic $k$-Hessian equations in $ ^N$.

Findings

Existence of solutions for certain parameter ranges.

Development of a fixed point theorem tailored to this problem.

Refinement of the critical Sobolev embedding.

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.