Suppose that $\Omega \subset \mathbb{R}^d$ is bounded with a Lipschitz boundary and $f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$. Show that, for a subsequence, $\nabla(f_j g_j) \rightarrow \nabla(fg)$ as a distribution. Find all $p \in [1,\infty]$ such that the convergence can be taken weakly in $L^p(\Omega)$.
Solution
Since $f_j \rightharpoonup f \in L^2$ and $\nabla f_j \rightharpoonup \nabla f \in L^2$, and similarly for $g_j$, then by Rellich-Kondrachov theorem, there exists $f_k:=f_{j_k} \to f \in L^2$ and $g_k:=g_{j_k} \to g \in L^2$, also $\{f_k\}$ and $\{g_k\}$ are bounded.
Let $\phi \in \mathcal{D}(\Omega)$,
\begin{aligned} \int_\Omega \left( \nabla (f_k g_k-fg) \right) \phi - &= \int_\Omega \left( f_k \nabla g_k - f_k \nabla g + f_k \nabla g - f \nabla g \right) \phi + \int_\Omega \left( g_k \nabla f_k -g_k \nabla f + g_k \nabla f - g \nabla f \right) \phi\\ & \leq \|\phi\|_{L^\infty}\left( \int_{\Omega} f_k (\nabla g_k -\nabla g) + \int_{\Omega} \nabla g(f_k-f) + \int_{\Omega} g_k (\nabla f_k -\nabla f) + \int_{\Omega} (g_k-g) \nabla f\right) \to 0.\\ \end{aligned}
Where the above follows by weak convergence.
The final part of the question is not clear to me. I tried to use the Sobolev embedding theorem but the gradient and the product confused me. Could you please help me! Thanks in advance.