In the above equations we used the explicit lattice-gas expressions for the entropy (18). Equations for Δ¯ µ and Δ¯ µ$_{c}$ follow from the bulk EL equations. The boundary conditions of the EL equations for the functional (26) are and where J is defined in (35). When ϵ = ¯ ϵ , the third EL simplifies, Alternatively, ψ could be eliminated, and three EL equations for the fields ϑ, η and φ would be obtained. The nonlinear EL equations (41)-(47) can be solved numerically. Since in the critical region and for weak surface fields and small σ the magnitudes of the fields φ , ϑ and η are small, we can obtain approximate analytical solutions within the perturbation method. Analytical solutions can give more general insight, but in the future studies should be supplemented with numerical solutions for particular choices of model parameters. We postulate that the solutions of the EL equations can be written in the form where f = ϑ, η, φ, ψ , f ( n ) = O ( ν $^{n}$), and ν is a small parameter. 15 ¯ ϵ + ϵ$_{ϑ}$ϑ (0) − ϵ$_{η}$η (0) 4 π ∇ ψ ( z ) |$_{z}$$_{=0}$ = − eσ (46) g $^{′′}$( z ) − 2 g $^{′}$g ( z )( e/k$_{B}$T ) + (2 πe/ ¯ ϵ )( e/k$_{B}$T ) $^{2}$ρ$_{c}$ ( z ) g ( z ) $^{2}$(1 − g ( z ) $^{2}$) = 0 . (49) f = N ∑ n =1 f ( n ) (50) III. APPROXIMATE SOLUTIONS OF THE EL EQUATIONS H$_{ϑ}$ = ¯ h$_{ρ}$J$_{ρs}$ − ¯ h$_{s}$J$_{ρρ}$ J = ¯ s + h$_{ρ}$J$_{ρs}$ − h$_{s}$J$_{ρρ}$ J , H$_{η}$ = ¯ h$_{s}$J$_{ρs}$ − ¯ h$_{ρ}$J$_{ss}$ J = ¯ ρ$_{c}$ + h$_{s}$J$_{ρs}$ − h$_{ρ}$J$_{ss}$ J . (48) ϑ ′ − ϑ = H$_{ϑ}$, η ′ − η = H$_{η}$, (47)