Yamabe-type Equations on Complete, Noncompact Manifolds

The aim of this monograph is to present a self-contained introduction to some geometric and analytic aspects of the Yamabe problem, and to describe in a way accessible to non-specialists a range of methods and techniques that can be successfully applied to nonlinear differential equations in situations where the lack of compactness and of symmetry and homogeneity prevents the use of more standard tools typical of compact situations or of the Euclidean setting. After providing a self-contained treatment of the geometric tools used in the book, the reader is introduced to the main subject through a concise but clear study of some aspects of the Yamabe problem on compact manifolds, which provides motivation and geometrical feeling for the subsequent study. In the main body of the book it is shown how the geometry and the analysis of nonlinear partial differential equations blend together to give up to date results on existence, nonexistence, uniqueness and a priori estimates for solutions of general Yamabe-type equations and inequalities on complete, non-compact Riemannian manifolds.