### A compactness result for polyharmonic maps in the critical dimension

For $n=2m\ge 4$, let $\Omega \in {\mathbb{R}}^{n}$ be a bounded smooth domain and $\mathcal{N}\subset {\mathbb{R}}^{L}$ a compact smooth Riemannian manifold without boundary. Suppose that $\left\{{u}_{k}\right\}\in {W}^{m,2}(\Omega ,\mathcal{N})$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}{E}_{m}\left(\Pi (u+t\xi )\right)=0$$ with ${\Phi}_{k}\to 0$ in ${\left({W}^{m,2}(\Omega ,\mathcal{N})\right)}^{*}$ and ${u}_{k}\rightharpoonup u$ weakly in ${W}^{m,2}(\Omega ,\mathcal{N})$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-${W}^{m,2}$ topology.