Minimizing non-convex multiple integrals: a density result

We consider variational problems of the form min{integral(Ohm)[f(del u) + g(u)] dx : u is an element of u(0) + W-0(1,p) (Ohm)}, (P) where Ohm is a bounded open set in R-N, f : R-N --> R is a possibly non-convex lower semicontinuous function with p-growth at infinity for some 1 < p < infinity, and the boundary datum u(0) is any function in W-1,W-p(Ohm). Assuming that the convex envelope f** of f is affine on each connected component of the set {f** < f}, we prove the existence of solutions to (P) for every continuous function g such that (i) g has no strict local minima and (ii) every convergent sequence of extremum points of g eventually belongs to an interval where g is constant, thus showing that the sei of continuous functions g that yield existence to (p) is dense in the space of continuous functions on R.