  Capillarity functionals are parameter invariant functionals defined on
classes of two-dimensionals parametric surfaces in R3 as the sum of the area
integral with an anisotropic term of suitable form. In the class of parametric
surfaces with the topological type of S2 and with fixed volume, extremals of
capillarity functionals are surfaces whose mean curvature is prescribed up to a
constant. For a certain class of anisotropies vanishing at infinity, we prove
existence and nonexistence of volume- constrained, S2-type, minimal surfaces
for the corresponding capillarity functionals. Moreover, in some cases, we show
existence of extremals for the full isoperimetric inequality.
