  We proceed further with the study of minimum weak Riesz energy problems for
condensers with touching plates, initiated jointly with Bent Fuglede (Potential
Anal. 51 (2019), 197--217). Having now added to the analysis constraint and
external source of energy, we obtain a Gauss type problem, but with weak energy
involved. We establish sufficient and/or necessary conditions for the existence
of solutions to the problem and describe their potentials. Treating the
solution as a function of the condenser and the constraint, we prove its
continuity relative to the vague topology and the topologies determined by the
weak and standard energy norms. We show that the criteria for the solvability
thus obtained fail in general once the problem is reformulated in the setting
of standard energy, thereby justifying an advantage of weak energy when dealing
with condensers with touching plates.
