I am given the problem $$ \min_{u \in H_0^1([0, 1])} \int^1_0 xu'(x)^2~\mathrm{d}x $$ and supposed to prove that the problem has no minimizer. But the integrand is bounded below by zero, and thus so is the functional that we want to minimize. The zero function is admissible and therefore $0$ is a minimizer.
Where is my mistake? Or is there a problem with the task?
Yes, you are right. Probably, the constraints should be $u \in H^1([0,1])$ with $u(0) = 1$, $u(1) = 0$. Then, the problem fails to possess a solution.