Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow L^{q}(\Omega)$$ to be (weak $\times$ norm, norm) continuous. Does the following argument make sense: If $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$ then it follows that $\nabla u \rightharpoonup \nabla u $ in $L^{p}(\Omega;\mathbb{R}^{n})$. Therefore $$f(u_{k},\nabla u_{k}) \rightharpoonup f(u,\nabla u)~~\text{in }L^{q}(\Omega)$$
Is this fine?
If this is not fine, can it at least be shown that $\{f(u_{k}, \nabla u_{k})\}_{k \in \mathbb{N}}$ is bounded in $L^{q}(\Omega)$?
Thanks for any help.