We explore the pressure versus temperature phase diagram of a system of dimeric Janus nanoparticles using molecular dynamics simulations. Each nanoparticle is modeled as a dumbbell which has one monomer that interacts by a standard Lennard-Jones potential while the other monomer interacts by a core-softened potential. The systems composed by particles interacting only by core-softened potential exhibit the density and the diffusion anomalous behavior observed in water while if the particles interact only by the Lennard-Jones potential no anomaly is present. Here we explore if the anomalous behavior is present when half of the particles are modeled by a core-softened potential and half with Lennard-Jones potential. We show that the diffusion anomaly is preserve, while the density anomaly can disappear depending on the nonanomalous monomer characteristics. We also show that the self-assembly structures characteristics of the dumbbell systems are affected by the balance between core-softened and non-core-softened monomers.