The paper is devoted to determining necessary and sufficient conditions for existence of solutions to the problem inf{ess sup(x is an element of Omega) f(del u(x)) : u is an element of u(0) + W-0(1,infinity)(Omega)}, when the supremand f is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Several conditions of convexity for the supremand f are also investigated.