Radial biharmonic $k-$Hessian equations: The critical dimension

TL;DR

This paper investigates radial solutions to a biharmonic $k$-Hessian elliptic PDE in a unit ball, revealing a critical dimension at N=4 for the case k=2, and explores existence of solutions with decay at infinity.

Contribution

It provides new insights into the existence and behavior of solutions to biharmonic $k$-Hessian equations, highlighting the critical dimension for $k=2$ and analyzing boundary conditions.

Findings

Dimension N=4 is critical for $k=2$ solutions.

Existence of solutions depends on boundary conditions and data.

Solutions decay at infinity under certain conditions.

Abstract

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.