Existence of Scalar Minimizers for Nonconvex Simple Integrals of Sum Type

One usually asks, as sufficient condition to guarantee the existence of minimizers x:wa, bxªR in AC for the integral Hab h x9.qw x.4dt}with h ?. having superlinear growth at ` and w ?. being any lower semicontinuous function bounded below}that h ?. be convex, that is, h j .sh** j . at all j in R. The purpose of this paper is to show that the condition h 0.sh** 0. is actually enough to guarantee the existence of minimizers.