We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in W 2,p . The result is improved for p = 2 avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón-Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori-Yau maximum principle for the Hessian. ContentsProblem 1.1. Under which (geometric) assumptions on M does one have that W 2,p 0 (M ) = W 2,p (M )? Classical results on this topic can be found in [6], [24] and references therein. In the following proposition we collect the most up-to-date achievements: point (I) was shown by E. Hebey, [23, Theorem 2.8]; point (II) was proved by B. Güneysu in [18, Proposition III.18]; point (III) is due to L. Bandara, [7] (for an alternative proof see also [18, Proposition III.18]). Proposition 1.2. Let (M m , g) be a complete Riemannian manifold.

Abstract. We prove global comparison results for the p-Laplacian on a p-parabolic manifold. These involve both real-valued and vector-valued maps with finite p-energy.

We investigate the existence of a solution and stability issues for the Einstein-scalar field Lichnerowicz equation in closed 3-manifolds in the framework of the Einstein-Maxwell theory. The results we obtain provide a complete picture for both the questions of existence and stability.

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