Solutions of elliptic equations on manifolds with roughly Euclidean ends (Q1346334)

A Riemannian manifold \(M\) is called ``with finitely many ends'' if the number of unbounded components of \(M\backslash C\) has a finite upper bound for any choice of a compact set \(C\). A function \(f\) is called \({\mathcal A}\)-harmonic if it satisfies an equation \(\text{div }{\mathcal A}(\nabla u)= 0\), where \({\mathcal A}: TM\to TM\) satisfies \(\langle{\mathcal A}(\nabla u), \nabla u\rangle\sim |\nabla u|^p\), with \(1< p< \infty\). It is also assumed that the injectivity radius of \(M\) is positive, and that the Ricci curvature has a lower bound. Finally, the author introduces the notion of parabolic and of hyperbolic ends. The main result of the paper studies Liouville-type theorems, i.e. it is analyzed under what conditions positive or bounded \({\mathcal A}\)-harmonic functions will exist. (The manifold is assumed complete.) An example is the following: \(M\) admits non-constant positive \({\mathcal A}\)-harmonic functions but no bounded \({\mathcal A}\)-harmonic functions except constants, if it has exactly one \(p\)-hyperbolic and at least one \(p\)-parabolic end.