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

Abstract

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$, and $k\ge 1$ an integer such that $n>2k$. We show that there exists $B_0>0$ such that for all $u\in H^k(M)$, \[\|u\|_{L^{2^\sharp}(M)}^2 \leq K_0^2 \int_M |Δ_g^{k/2} u|^2 \,dv_g + B_0 \|u\|_{H^{k-1}(M)}^2,\] where $2^{♯}=\frac{2n}{n-2k}$ and ${\Delta }_g=-{\mathrm{div}}_g(\nabla \cdot )$. Here $K_0$ is the optimal constant for the Euclidean Sobolev inequality $({\int }_{{\mathbb{R}}^n}|u{|}^{2^{♯}}{)}^{2/2^{♯}}\le K_0^2{\int }_{{\mathbb{R}}^n}|{\nabla }^{k}u{|}^2$ 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 $({\Delta }_g+\alpha {)}^{k}u=u^{2^{♯}-1}$ in $M$. We obtain a pointwise description of $u_{\alpha }$, with explicit dependence in $\alpha$ as $\alpha \to \infty$.

57 pages, comments welcome