On p-harmonic maps and convex functions

We prove that, in general, given a p-harmonic map F : M → N and a convex function ${H : N \rightarrow \mathbb{R}}$ , the composition ${H\circ F}$ is not p-subharmonic, if p ≠ 2. This answers in the negative an open question arisen from a paper by Lin and Wei. By assuming some rotational symmetry on manifolds and functions, we reduce the problem to an ordinary differential inequality. The key of the proof is an asymptotic estimate for the p-harmonic map under suitable assumptions on the manifolds.