Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand

Let there be given a non-negative, quasiconvex function F satisfying the growth condition lim supA→∞ F(A)/|A|p = 0 (*) for some p ∈ ] 1, ∞ [. For an open and bounded set Ω ⊂ ℝm, we show that if q≧ m-1/m p and q > 1, then the variational integral ℱ(u; Ω):= ∫Ω F(Du) dx is lower semicontinuous o...