Yamabe type equations with sign-changing nonlinearities on non-compact Riemannian manifolds

We study the existence problem for positive solutions of Yamabe type equations on complete manifolds on manifolds with a pole, the main novelty being that the coefficient of the nonlinearity is allowed to change sign. This class of PDEs arises in a number of different geometric situations, notably the prescribed scalar curvature problem, but the sign-changing case has remained basically unsolved in the literature, with the exception of few special cases. This paper aims at giving a unified treatment together with new, general existence theorems expressed in terms of the growth at infinity of the coefficient of the nonlinearity with respect to the geometry of the manifold. We prove that our results are sharp and that, even for Euclidean space, they improve on previous works in the literature. Furthermore, we also detect the asymptotic profile of the solution at infinity, and a detailed description of the influence of the linear term in the equation and of the geometry of the underlying manifold on this profile is given. The possibility to express the assumptions in an effective way also depends on some new asymptotic estimates for solutions of the linear Cauchy problem for a Sturm-Liouville type ODE, of independent interest.