We consider variational problems whose lagrangian is of the form f(Du)+g(u) where f is a possibly non-convex lower semicontinuous function with p-growth at infinity for some 1 < p < ∞, and the boundary datum is any function in W 1,p (Ω). Assuming that the convex envelope 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 set of continuous functions g that yield existence to (P) is dense in the space of continuous functions on R.