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

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 ${\phi }_a:B^n\to {\mathbb{S}}^n$, 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 $\mathit{\text{unstable}}$. In the course of this analysis we point out some specific differences between the $r$-harmonic and the $ES-r$-harmonic cases when $r\ge 4$. 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 $n\ge 5$, as well as proper weakly triharmonic maps for all $n\ge 7$. Finally, we study a similar problem replacing the domain $B^n$ with a suitable warped product manifold.