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

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