A uniqueness result for a translation invariant problem in the calculus of variations

We present a uniqueness result of uniformly continuous solutions for a general minimization problem in the Calculus of Variations. We minimize the functional I λ (u) := Ω φ(∇u)+λu with φ a convex but not necessarily strictly convex function, Ω an open set of R N with N ∈ N * and λ ∈ R. The proof is based on the two following main points : the functional I λ is invariant under translations and we assume that the function φ is not affine on any non-empty open set. This provides a shorter proof and/or an extension for some already known uniqueness results for functionals of the type I λ that are presented in the article