Condenser capacities and capacitary potentials for unbounded sets, and global <i>p</i> -harmonic Green functions on metric spaces

We study the condenser capacity cap_<i>p</i>⁡(<i>E</i>, Ω) on <i>unbounded</i> open sets Ω in a proper connected metric space <i>X</i> equipped with a locally doubling measure supporting a local <i>p</i>-Poincaré inequality, where 1&lt;p&lt;∞. Using a new definition of capacitary potentials, we show that cap_<i>p</i> is countably subadditive and that it is a Choquet capacity. We next obtain formulas for the capacity of superlevel sets for the capacitary potential. These are then used to show that <i>p</i>-harmonic Green functions exist in an unbounded domain Ω if and only if either <i>X</i> is <i>p</i>-hyperbolic or the Sobolev capacity Cp(X∖Ω)&gt;0. As an application, we deduce new results for Perron solutions and boundary regularity for the Dirichlet boundary value problem for <i>p</i>-harmonic functions in unbounded open sets.