Existence and stability of weak critical points of $r$-energy functionals

TL;DR

This paper proves the existence and instability of certain weakly r-harmonic maps between specific geometric domains and targets, exploring their properties across different dimensions and variants.

Contribution

It constructs explicit weakly r-harmonic maps in symmetric settings, analyzes their stability, and extends results to ellipsoids and warped product domains.

Findings

Existence of weakly r-harmonic maps depends on dimension n.

Constructed critical points are shown to be unstable.

Extended analysis to ellipsoids and warped product manifolds.

Abstract

The main aim of this paper is to prove the existence of certain proper weakly $r$-harmonic ($ES-r$-harmonic) maps. We construct critical points which belong to a family of rotationally symmetric maps $\varphi_a : B^n \to \mathbb{S}^n$, where $B^n$ and $\mathbb{S}^n$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. We find that the existence of solutions within this family is restricted to specific dimensions $n$. Next, we prove that our critical points are \textit{unstable}. In the course of this analysis we point out some specific differences between the $r$-harmonic and the $ES-r$-harmonic cases when $r \geq 4$. Next, we analyse two variants of the problem. First, we replace the target manifold $\mathbb{S}^n$ with a rotationally symmetric ellipsoid $E^n(b)$ and establish the existence of proper weakly biharmonic maps for all $n \geq 5$, as well as proper weakly triharmonic maps for all $n \geq 7$. Finally, we study a similar problem replacing the domain $B^n$ with a suitable warped product manifold.