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Motivated by thermodynamic analogy of black holes and Van der Waalsliquid/gas system, in this paper, we study $P-V$ criticality of both dilatonicBorn-Infeld black holes and their conformal solutions, Brans-Dicke-Born-Infeldsolutions. Due to the conformal constraint, we have to neglect the oldLagrangian of dilatonic Born-Infeld theory and its black hole solutions, andintroduce a new one. We obtain spherically symmetric nonlinearly charged blackhole solutions in both Einstein and Jordan frames and then, we calculate therelated conserved and thermodynamic quantities. After that, we extend the phasespace by considering the proportionality of the cosmological constant andthermodynamical pressure. We obtain critical values of thermodynamiccoordinates through numerical methods and plot relevant $P-V$ and $G-T$diagrams. Investigation of the mentioned diagrams helps us to studythermodynamical phase transition. We also analyze the effects of varyingdifferent parameters on the phase transition of black holes.

Extended phase space thermodynamics and $P-V$ criticality: Brans-Dicke-Born-Infeld vs Einstein-Born-Infeld-dilaton black holes