= Solution
The integrand is nonnegative. Define the <Lipschitz> triangular waves
$$
u_N(x)=2\operatorname{dist}(x,N^{-1}\mathbb Z),\qquad0\leq x\leq1.
$$
They have zero endpoint values, slope $\pm2$ <almost everywhere>, and height $1/N$. Hence $(u_N'^4-16)^4=0$ <almost everywhere>. On each of the $N$ equal cells, direct integration of the fourth power of the two affine pieces gives $1/(5N^5)$. Therefore
$$
\boxed{I(u_N)=\frac1{5N^4}\longrightarrow0,\qquad\inf_{\mathcal A}I=0.}
$$
This is an <oscillating minimizing sequence with no minimizer>.
\Image[/past-exam-of-the-calculus-of-variations-course-of-the-university-of-oxford/2009-oscillating-minimizing-sequence.png]
{title=Oscillations keep the derivative at the two wells while the amplitude vanishes}
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