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We compute models for the equation of state (EoS) of the matter in the coresof hybrid stars. Hadronic matter is treated in the non-linear relativisticmean-field approximation, and quark matter is modeled by three-flavor local andnon-local Nambu$-$Jona-Lasinio (NJL) models with repulsive vector interactions.The transition from hadronic to quark matter is constructed by consideringeither a soft phase transition (Gibbs construction) or a sharp phase transition(Maxwell construction). We find that high-mass neutron stars with masses up to$2.1-2.4 M_\odot$ may contain a mixed phase with hadrons and quarks in theircores, if global charge conservation is imposed via the Gibbs conditions.However, if the Maxwell conditions is considered, the appearance of a purequark matter core either destabilizes the star immediately (commonly fornon-local NJL models) or leads to a very short hybrid star branch in themass-radius relation (generally for local NJL models).

Hybrid Stars in the Framework of different NJL Models