Strongly \(p\)-subharmonic functions and volume growth property of complete Riemannian manifolds (Q5956131)

A smooth function \(u\) on a non-compact complete Riemannian manifold \((M,g)\) is said to be \(p\)-subharmonic \((p>1)\) if \(\Delta_p u:=\text{div}_g (|\nabla u|^{p-2} \nabla u)\geq 0\). It is known that existence of a non-constant bounded \(p\)-subharmonic function on \((M,g)\) is closely related to the volume growth of the manifold. The author obtains lower bounds for geodesic balls in terms of the properties of a \(p\)-subharmonic (with \(p\geq 2)\) function \(u\) on the manifold and proves a generalized maximum principle for the \(p\)-Laplacian \(\Delta p\). This extends the author's results in [Nagoya Math. J. 151, 25-36 (1998; Zbl 0922.31009)] for the case \(p=2\) (i.e., for subharmonic functions) to the case \(p\geq 2\).