John E. BravoJean C. Cortissoz2026-03-222026-03-22202510.1017/s0004972725000061https://doi.org/10.1017/s0004972725000061https://andeanlibrary.org/handle/123456789/76552Abstract We show that March’s criterion for the existence of a bounded nonconstant harmonic function on a weak model (that is, $\mathbb {R}^n$ with a rotationally symmetric metric) is also a necessary and sufficient condition for the solvability of the Dirichlet problem at infinity on a family of metrics that generalise metrics with rotational symmetry on $\mathbb {R}^n$ . When the Dirichlet problem at infinity is not solvable, we prove some quantitative estimates on how fast a nonconstant harmonic function must grow.enMathematicsPure mathematicsarticle