In electrolyte solution study, people usually use a free energy formof an infinite domain system (with vanishing potential boundary condition) and the derived PDE(s) for analysis and computing.However, in many real systems and/or numerical computing, the objectivedomain is bounded, and people still use the similar energy form, PDE(s) but withdifferent boundary conditions, which may cause inconsistency.In this report, (1) we present a mean field free energy functional for electrolyte solution within a bounded domain witheither physical or numerically required artificial boundary. Apart from the conventional energy components,new boundary interaction terms are added for both Neumann and Dirichlet boundary conditions.(2) The traditional physical-based Poisson-Boltzmann (PB) equation and Poisson-Nernst-Planck (PNP) equations are proved to be consistent with the new free energy form, and different boundary conditions can be applied.(3) In particular, for inhomogeneous electrolyte with ionic concentration-dependent dielectric permittivity, we derive the generalized Boltzmann distribution (thereby the generalized PB equation) for equilibrium case, and the generalized PNP equations (VDPNP) for non-equilibrium case, under different boundary conditions.(4) Furthermore, the energy laws are calculated and compared to study the energy properties of differentenergy functionals and the resulted PNP systems.