On p-harmonic self-maps of spheres

Abstract

In this manuscript we study rotationally $p$-harmonic maps between spheres. We prove that for $p\in \mathbb{N}$ given, there exist infinitely many $p$-harmonic self-maps of ${\mathbb{S}}^m$ for each $m\in \mathbb{N}$ with $p<m<2+p+2\sqrt{p}$. In the case of the identity map of ${\mathbb{S}}^m$ we explicitly determine the spectrum of the corresponding Jacobi operator and show that for $p>m$, the identity map of ${\mathbb{S}}^m$ is stable when interpreted as a $p$-harmonic self-map of ${\mathbb{S}}^m$.