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This article develops optimality conditions for a large class of non-smoothvariational models. The main results are based on standard tools of functionalanalysis and calculus of variations. Firstly we address a model with equalityconstraints and, in a second step, a more general model with equality andinequality constraints, always in a general Banach space context. We highlightthe results in general are well known, however, some novelties are introducedrelated to the proof procedures, which are in general softer than thoseconcerning the present literature.

On Lagrange multiplier theorems for non-smooth optimization for a large class of variational models in Banach spaces