Integral functionals that are continuous with respect to the weak topology on W01,p(Ω)

Let Ω⊂RNΩ⊂RN be a bounded open set and let g:Ω×R→Rg:Ω×R→R be a Carathéodory function that satisfies standard growth conditions. Then the functional Φ(u)=∫Ωg(x,u(x))dx is weakly continuous on W01,p(Ω), 1≤p≤∞1≤p≤∞, if and only if gg is linear in the second variable.