Classification of Solutions to a Critical Fourth-Order Equation on Complete Non-compact Manifolds

We study the critical biharmonic equation $${\mathrm{\Delta }}^{2}u=u^{\frac{n+4}{n-4}},$$ on a complete, connected, and non-compact Riemannian manifold $(M^n,g)$ of dimension $n\ge 5$ with non-negative Ricci curvature. We establish an optimal pointwise second-order derivative estimate by Bernstein’s method and the continuity method. Using invariant tensors, we derive a differential inequality that yields a rigidity result. More precisely, if there exists a positive finite energy solution, then the manifold is isometric to the Euclidean space ${\mathbb{R}}^n$, and the solution is given by $$u(x)=\frac{{[{\lambda }^{2}n(n-4)(n^2-4)]}^{\frac{n-4}{8}}}{{(1+\lambda \mathrm{\mid }x-x_0{\mathrm{\mid }}^2)}^{\frac{n-4}{2}}},\lambda >0,x_0\in {\mathbb{R}}^n\mathrm{.}$$