Existence of nontrival $n$-harmonic maps via min-max methods

For any $n\ge 3$ and any closed manifold $\mathcal{N}$ with ${\pi }_{n+k}(\mathcal{N})\mathrm{\ne }\{0\}$ for some $k\ge 0$, we establish the existence of nontrivial $n$-harmonic maps from ${\mathbb{S}}^n$ into $\mathcal{N}$. When $k\ge 1$, these maps naturally appear as bubbling limits of $p$-harmonic maps with $p>n$, obtained by min-max constructions in the limit $p\to n^{+}$.