Solution
= 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 $|u'|=2$ <almost everywhere>. Thus
$$
\boxed{\text{The infimum zero is not attained}.}
$$