Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold M, namely, the existence of a conformal deformation of the metric realizing a given function as its scalar curvature. In particular, the work focuses on the case when the prescribed scalar curvature changes sign. Our main achievements are two existence results requiring minimal assumptions on the underlying manifold, and ensuring a control on the stretching factor of the conformal deformation in such a way that the conformally deformed metric be bi-Lipschitz equivalent to the original one. The topological-geometrical requirements we need are all encoded in the spectral properties of the standard and conformal Laplacians of M. Our techniques are extended to investigate the existence of entire positive solutions of quasilinear equations of p-Laplacian type, and in the process of collecting the background material, we give some new insight on the subcriticality theory for the Schr"odinger type operator describing the homogeneous part of the equation. In particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.