Subharmonic functions on graphs (Q1366850)

Liouville type theorems on infinite connected graphs with uniformly bounded vertex degree are studied parallel to the continuous case, where the geometry of a complete manifold is controlled in terms of the volume only. Along the lines of \textit{L. Karp} [Math. Z. 179, 535-554 (1982; Zbl 0441.31005)] with nontrivial adaptations due to the discrete setup Theorem A is proved: Let \(u\) be a non-negative subharmonic function on the graph \(G\). Then either \(u\) is constant or, for any \(2\leq p < +\infty\), \(\lim_{R\to +\infty} R^{-2} \sum_{x\in B_R(q)} u^p(x) = +\infty\), where \(q\) is any vertex of \(G\) and \(B_R(q)\) the ball of radius \(R\) around \(q\) in the shortest path metric. Partial results in the same direction are obtained in the case \(1 < p < 2\). Inspired by \textit{P. Li} and \textit{L.-F. Tam} [J. Differ. Geom. 29, 421-425 (1989; Zbl 0668.53023)] logarithmic growth constraints on subharmonic functions are considered resulting in Theorem E: Suppose that, for some vertex \(q\), \(\lim_{R\to +\infty} |B_R(q)|R^{-2} = 0\). Let \(u\) be a subharmonic function such that \(u(x) \leq A \log \rho(x,q) + B\) for some constants \(A, B > 0\), the shortest path distance \(\rho(x,q)\) and \(|\cdot|\) denoting the cardinality. Then \(u\) is constant. This theorem seems to be new, even in the continuous case.