Attaining the optimal constant for higher-order Sobolev inequalities on manifolds via asymptotic analysis

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\geq 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u \in H^{k}(M)$, $$\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |\Delta_g^{k/2} u|^2 \,dv_g + B_0 \|u\|_{H^{k-1}(M)}^2,$$ where $2^\sharp = \frac{2n}{n-2k}$ and $\Delta_g = -\operatorname{div}_g(\nabla\cdot)$. Here $K_0$ is the optimal constant for the Euclidean Sobolev inequality $\big(\int_{\mathbb{R}^n} |u|^{2^\sharp}\big)^{2/2^\sharp} \leq K_0^2 \int_{\mathbb{R}^n} |\nabla^k u|^2$ for all $u \in C_c^\infty(\mathbb{R}^n)$. This result is proved as a consequence of the pointwise blow-up analysis for a sequence of positive solutions $(u_\alpha)_\alpha$ to polyharmonic critical non-linear equations of the form $(\Delta_g + \alpha)^k u = u^{2^\sharp-1}$ in $M$. We obtain a pointwise description of $u_\alpha$, with explicit dependence in $\alpha$ as $\alpha\to \infty$.