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

Abstract

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 s(x) as its scalar curvature. In particular, the work focuses on the case when s(x) changes sign. Our main achievement are two new 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 can be extended to investigate the existence of entire positive solutions of quasilinear equations of the typewhere ∆p is the p-Laplacian, σ > p − 1 > 0, a, b ∈ L ∞ loc (M ) and b changes sign, and in the process of collecting the material for the proof of our theorems, we have the opportunity to give some new insight on the subcriticality theory for the Schrödinger type operatorIn particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.

ImprovedLp-Poincaré inequalities on the hyperbolic space

“…We recall for the convenience of the reader the proof given in [7], only the asymptotics at infinity not being explicitly given there. The proof relies on the well known classical Hardy inequality with respect to the Green's function and exploiting its behavior on hyperbolic space.…”

“…The proof relies on the well known classical Hardy inequality with respect to the Green's function and exploiting its behavior on hyperbolic space. More precisely, for N ≥ 2 and p > 1, the following Hardy inequality holds (see [13], [7]): The proof is then a calculus exercise involving the asymptotics of the function G p (r). Indeed, Eq.…”

Lichnerowicz-type equations on complete manifolds

“…In particular, Corollary 3.12 is our main Liouville-type theorem. In Section 4 we analyze another uniqueness result, this time under a spectral assumption on the manifold, in the spirit of the very recent [5] and [6].…”

A note on Killing fields and CMC hypersurfaces

“…The critical curve, and its relationship with Green kernels, appears in a number of geometric problems, among them Yamabe-type equations on complete non-compact manifold (see [5,3]). The interested reader is suggested to consult [4] for deepening.…”