Bounded variation solutions of capillarity-type equations

We investigate by different techniques, the solvability of the capillarity-type problem −div ( ∇u/ √ 1 + |∇u| ) = f(x, u) in Ω, −∇u · n/ √ 1 + |∇u| = κ(x) on ∂Ω, (1) where Ω is a bounded domain in RN , f : Ω × R → R is a Caratheodory function, n is the unit outer normal to ∂Ω and κ : ∂Ω → R is a bounded function. Since our approach is variational, the natural context where this problem has to be settled is the spaceBV (Ω) of bounded variation functions. Solutions of (1) are defined as subcritical points of the action functional ∫