Entire $p$-harmonic functions with an isolated critical point and failure of $C^1$-regularity of the natural gradient

For every $n\ge2$ and $1<p<\infty$, we show an existence of a nonconstant homogeneous entire $p$-harmonic function whose only critical point is the origin. The angular parts of these functions are axially symmetric and even across the equator. To establish the solution we use contraction argument at