The Liouville theorem for $p$-harmonic functions and quasiminimizers with finite energy

We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite $p$th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global $p$-Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line $\mathbf{R}$, we characterize all locally doubling measures, supporting a local $p$-Poincaré inequality, for which there exist nonconstant quasiminimizers of finite $p$-energy, and show that a quasiminimizer is of finite $p$-energy if and only if it is bounded. As $p$-harmonic functions are quasiminimizers they are covered by these results.

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