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By means of the Royden compactification of an open Riemann surface  R  necessary and sufficient conditions are given for a Dirichlet-finite solution of      Δ  u = P u ( P ≧  0 , P ≢ 0 ) \Delta u = Pu\;(P \geqq 0, P\;{\nequiv }\;0)       to be  PD -minimal on  R . A relation between  PD -minimal solutions and  HD -minimal solutions is obtained. In addition it is shown that the dimension of the space of  PD -solutions is the same as the number of  P -energy nondensity points in the finite dimensional case.

𝑃𝐷-minimal solutions of Δ𝑢=𝑃𝑢 on open Riemann surfaces