Multilayer and multiplex networks represent a good proxy for the
description of social phenomena where social structure is important and can have
different origins. Here we propose a model of opinion competition where individuals
are organized according to two different structures in two layers. Agents exchange
opinions according to the Abrams-Strogatz model in each layer separately and opinions
can be copied across layers by the same individual. In each layer a different opinion
is dominant and then each layer has a different absorbing state. The consensus in one
opinion is not the only possible stable solution because of the interaction between the
two layers. A new mean field solution where both opinions coexist has been found.
In a finite system there is a long transient time of a dynamical coexistence of both
opinions, however, the system ends in a consensus state due to finite size effects. We
analyze sparse topologies in the two layers and the existence of positive correlations
between the layers that enables the coexistence of inter-layer groups of agents sharing
the same opinion.