A new sharp Sobolev inequality between the $Q$-curvature and the scalar curvature on Euclidean space

We establish a sharp Sobolev-type inequality on $\mathbb{R}^n$, $n\geq 5$, that interpolates between the second-order energy and a weighted first-order term. Specifically, we determine the sharp constant $S_{4,2}$ in the inequality \begin{equation*} \int_{\mathbb R^n} (Δu)^2\,dx \ge S_{4,2} \left(\int_{\mathbb{R}^n}|\nabla u|^{2}|u|^{\frac{4}{n-4}}\,dx\right)^{\frac{n-4}{n-2}}, \end{equation*} for any $u\in\dot H^2(\mathbb{R}^n)$, and we classify the cases of equality. This solves the problem of minimizing the total $Q$-curvature of among metrics on the sphere, conformal to the standard metric, with fixed total scalar curvature.