Say that $(u_n)_{n\geq 1}\subset W^{1,p}(\Omega)$ and $u\in W^{1,p}(\Omega)$ where $\Omega\subset\mathbb{R}^N$ is an open and bounded set and $\infty>p>1$. We know that:
$$u_n\to u\ \text{in}\ L^{p}(\Omega)$$
We also know that $\nabla u_n$ is bounded in $L^{p}(\Omega)^N$. Can we infer from these facts that:
$$\nabla u_n\rightharpoonup \nabla u\ \text{in}\ L^{p}(\Omega)^N$$?
Of course, since $L^{p}(\Omega)$ is reflexive, there is some $\mathbf{v}\in L^{p}(\Omega)^N$ (from Eberlein-Smulian theorem) such that (on a subsequence denoted the same way):
$$\nabla u_n\rightharpoonup \mathbf{v}\ \text{in}\ L^{p}(\Omega)^N$$
How can we prove that $\mathbf{v}=\nabla u$?
