How to solve this variational problem related to the Couette Flow?

Here is a proposal, how this could be achieved. I have no idea whether these arguments work in the cylindrical coordinates setting.

First, take a minimizing sequence $(u_n)$. Since $\inf J<0$, let me assume $J(u_n)<0$ and $u_n\ne0$ for all $n$. Then we can normalize the sequence to have $\|\nabla u_n\|_{L^2}=1$. Hence the sequence is bounded in $H^1$. Then (after extracting a subsequence) if necessary, we have $u_n \rightharpoonup u$ in $H^1$ and (due to compact embeddings) $u_n \to u$ in $L^2$. This should be enough to pass to the limit in numerator, $$ \int_\Omega r^{-4} u_{n,\theta}u_{n,r} \to \int_\Omega r^{-4} u_{\theta}u_{r} \le0. $$ In addition, $X$ should be a closed subspace, so $u\in X$. And we should be able to pass to the lim-inf in the functional: since the numerators along the sequence are negative we have $$ \liminf J(u_n) = \left(\lim \int_\Omega r^{-4} u_{n,\theta}u_{n,r}\right)\cdot \limsup (\|\nabla u_n\|_{L^2}^{-2}) = \left(\int_\Omega r^{-4} u_{\theta}u_{r} \right)\cdot ( \liminf \|\nabla u_n\|_{L^2})^{-2} \ge \left(\int_\Omega r^{-4} u_{\theta}u_{r} \right)( \|\nabla u\|_{L^2})^{-2} = J(u). $$ (This inequality shows that $t \mapsto - f(t)^{-2}$ is lower-semicontinuous for lower semicontinuous and non-negative $f$.)