We propose two 3-3-1 models (with either neutral fermions or right-handed neutrinos) based on S 3 flavor symmetry responsible for fermion masses and mixings. The models can be distinguished upon the new charge embedding (L) relevant to lepton number. The neutrino small masses can be given via a cooperation of type I and type II seesaw mechanisms.The latest data on neutrino oscillation can be fitted provided that the flavor symmetry is broken via two different directions S 3 → Z 2 and S 3 → Z 3 (or equivalently in the sequel S 3 → Z 2 → {Identity}), in which the second direction is due to a scalar triplet and another antisextet as small perturbation. In addition, breaking of either lepton parity in the model with neutral fermions or lepton number in the model with right-handed neutrinos must be happened due to the L-violating scalar potential. The TeV seesaw scale can be naturally recognized in the former model. The degenerate masses of fermion pairs (µ, τ ), (c, t) and (s, b) are respectively separated due to the S 3 → Z 3 breaking.

New five-dimensional charged AdS black hole solutions are found in Einstein-Gauss-Bonnet gravity and the nonlinear electrodynamics. These solutions include regular black holes as well as extremal black holes. The first law of the black hole thermodynamics is confirmed in the extended phase space where the cosmological constant is treated as the pressure. The first and second order phase transitions are investigated by observing the behavior of the heat capacity at constant pressure and the Gibbs free energy. In addition, the equation of state for the black holes and their P −V criticality are studied. Finally, the critical exponents are found to be the same as those of the Van der Waals fluid.

In the context of the non-linear electrodynamics and the Einstein-massive gravity, we have obtained a 4D non-linear charged AdS black hole solution. Then, we investigated its horizon structure. In addition, the thermodynamics and phase structure of this black hole solution have been studied in details. We have computed various thermodynamic quantities of the black hole, such as the temperature, entropy, the heat capacity at constant pressure, or the Gibbs free energy. The black hole can undergo the first-order, second-order phase transitions which depend crucially on the effective horizon curvature, the sign of the coupling parameter c 1 , the characteristic parameter of the non-linear electrodynamics, as well as the pressure. Finally, we derived the equation of state and studied P − V criticality in the case of the positive effective horizon curvature.

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