Critical points for multiple integrals of the calculus of variations

In this paper we deal with the existence of critical points of functionals defined on the Sobolev space W-0(1,p)(Omega), p > 1, by J(u) = (Omega)integral S(x, u, Du) dx, where Omega is a bounded, open subset of R(N). Even for very simple examples in R(N) the differentiability of J(u) can fail. To overcome this difficulty we prove a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem applicable to functionals which are not differentiable in all directions. Existence and multiplicity of nonnegative critical points are also studied through the use of this theorem.