 Solution

If an admissible $u$ attained the infimum, nonnegativity and $I(u)=0$ would force $u^4=0$ almost everywhere, so its continuous Sobolev representative would be identically zero. Its derivative would then vanish almost everywhere, but the other integrand term would equal $16^4>0$. This is impossible. Equivalently, zero energy would simultaneously require $u=0$ and $|u^{\prime }|=2$ almost everywhere. Thus