Instability as p-harmonic maps for a family of examples

The radial map $u\left(x\right)=\frac{x}{\Vert x\Vert }$ is a well-known example of a harmonic map from $R^m-\left\{0\right\}$ into the spheres $S^{m-1}$ with a point singularity at $x=$ 0. In Nakauchi (2023) the author constructed, for any positive integers $m$, $n$ satisfying $n\le m$, a family of harmonic maps $u^{\left(n\right)}$ from $R^m-\left\{0\right\}$ into the sphere $S^{m^n-1}$ with a point singularity at the origin, such that $u^{\left(1\right)}$ is the above radial map. It is known that for $m$ $\ge$ 3, the radial map $u^{\left(1\right)}$ is not only stable as a harmonic map but also a minimizer of the energy of harmonic maps. On the other hand in Nakauchi (2024) the author prove that for $n\ge 2$, the map $u^{\left(n\right)}$ is unstable if $m\ge 3$ and $n$ $>\frac{\sqrt{3}-1}{2}(m-1)$. It is remarkable that $u^{\left(n\right)}$ may be unstable in the case of $n$ $\ge$ 2.

We see that $u^{\left(n\right)}$ is a $p$  - harmonic map for any $p$ $>$ 0. In this paper we study the stability as a $p$  - harmonic map for $u^{\left(n\right)}$. The radial map $u^{\left(1\right)}$ is stable as a $p$  - harmonic map and furthermore it is a minimizing $p$  - harmonic map for any real number $p$ satisfying 1 $<p$ $<m$ (Coron and Gulliver, 1989; Hardt et al., 1998; Hong, 2001). We prove that for $n$ $\ge$ 2, the map $u^{\left(n\right)}$ is unstable as a $p$  - harmonic map if $m$ $>p$ $\ge$ 2 and $n$ $\ge$ $\frac{1}{2}\frac{m-p}{m-2}(m-p+1)$. It is also notable that for $n$ $\ge$ 2, the map $u^{\left(n\right)}$ may be unstable as a $p$  - harmonic map. Our results give many examples of unstable $p$  - harmonic maps into the spheres with a point singularity at the origin.