Relaxation of multiple integrals below the growth exponent

The integral representation of the relaxed energies (Fq,p)(u, Omega) := inf({un}){ lim inf(n -->infinity) integral(Omega) F(x, u(n), del u(n)) dx :u(n) is an element of W-1,W-q(Omega,R-d), u(n) --> u weakly in W-1,W-p(Omega, R-d)}, (F)(q,p)(loc)(u, Omega) := inf({un}){lim inf(n-->infinity) integral(Omega) F(x,u(n),del u(n))dx : u(n) is an element of W-loc(1,q)(Omega, R-d), u(n) --> weakly in W-1,W-p(Omega, R-d)} of a functional E : u bar right arrow integral(Omega) F(x,u,del u) dx, u is an element of W-1,W-q(Omega, R-d), where 0 less than or equal to F(x,zeta,xi) less than or equal to C(1 + \zeta\(r) + \xi\(q)) and max{1,rN-1/N+r,qN-1/N} < p less than or equal to q, is studied. In particular, W-1,W-p-sequentail weak lower semicontinuity of E(.) is obtained in the case where F = F(xi) is a quaiconvex function and p > q(N-1)/N.