Existence of minimizers for nonconvex variational problems with slow growth

Consider the minimization problem (P) min{integral(0)(1)f(t,u'(t)) dt; u is an element of W-1,W-1([0, 1], R-n), u(0) = u(0), u(1) = u(1)}, in which f:[0, 1] x R-n --> R boolean OR {+infinity} is a normal integrand. Define the convex function G: R-n --> R boolean OR {+infinity} by G(p) = integral(0)(1) f*(t,p) dt. It is known that, if the essential domain H of G is open, then problem (P) has a minimizer for any pair of endpoints (u(0), u(1)). In this paper, the same result is proved under the condition that, for every point p in H, the subgradient set partial derivative G(p) is either bounded or empty (when H is open, this condition herds automatically).