On a class of rotationally symmetric $p$-harmonic maps

We give a classification of rotationally symmetric \begin{document} $p$ \end{document} -harmonic maps between some model spaces such as \begin{document} $\mathbb{R}^n$ \end{document} and \begin{document} $\mathbb{H}^n$ \end{document} by their asymptotic behaviors. Among others, we show that, when \begin{document} $p>2$ \end{document} and \begin{document} $n≥q 2$ \end{document} , all rotationally symmetric \begin{document} $p$ \end{document} -harmonic maps from \begin{document} $\mathbb{R}^n$ \end{document} to \begin{document} $\mathbb{H}^n$ \end{document} have to blow up at a finite point, while all rotationally symmetric \begin{document} $p$ \end{document} -harmonic maps from \begin{document} $\mathbb{H}^n$ \end{document} to \begin{document} $\mathbb{H}^n$ \end{document} observe the trichotomy property, i.e. the map \begin{document} $y$ \end{document} is the identity map, is bounded or blows up according as its initial value \begin{document} $y'(0)$ \end{document} is equal to, less than or greater than one. Our sharp estimates imply and improve a number of existence and non-existence results of certain \begin{document} $p$ \end{document} -harmonic maps on noncompact manifolds.