 Solution

If an admissible $u$ attained the infimum zero, each nonnegative term of its integrand would vanish almost everywhere. In particular $|u{|}^{2011}=0$ almost everywhere, so $u=0$ in its Sobolev space and its weak derivative is zero. But then the remaining term is $(-64{)}^2=4096$ almost everywhere, giving $I(u)=4096$, a contradiction. Thus the infimum is zero but no global minimizer exists. The oscillatory sequence has no convergence of derivatives compatible with substitution into this nonconvex energy.