Provided for a sequence $u_n$ we know
\begin{eqnarray} u_n \rightarrow u \ \ \ \text{strongly in } L^p(\Omega) \\ u_n \rightharpoonup u \ \ \ \text{weakly in } H^{1,p}(\Omega), \end{eqnarray}
Can we conclude something like
\begin{equation} |\nabla u_n|^q \rightharpoonup |\nabla u|^q \ \ \ \text{weakly in } L^{\frac{p}{q}}(\Omega) \end{equation}
for $q<p$? We obviously have weak convergence of $|\nabla u_n|^q$ to at least some unidentified limit and the strong $L^p$-convergence implies
\begin{equation} u_n^q \rightarrow u^q \ \ \ \text{in } L^{\frac{p}{q}}(\Omega) \end{equation}
but as far as I see this only gives us
\begin{equation} \nabla u_n^q \rightharpoonup \nabla u^q \ \ \ \text{in } L^{\frac{p}{q}}(\Omega). \end{equation}
Is what I am trying to show wrong?