Suppose a sequence $\{u_{j}\}$ is bounded in the Sobolev space $H^{2+\epsilon}(\Omega)$, for $\epsilon>0$, where $\Omega$ is say a bounded, $C^{\infty}$ domain. Here, the fractional space $H^{s}(\Omega)$ is defined as the restriction of elements of $H^{s}(\mathbb{R}^{n})$ to $\Omega$, but this definition is equivalent to the usual definition of $W^{s,q}(\Omega)$ by the extension property.
By weak* compactness, we have that there is a $u\in H^{2+\epsilon}(\Omega)$ such that (up to a subsequence) $u_{j}\rightharpoonup u$. By Rellich-Kondrachov, $u_{j}\rightarrow u$ in $H^{2}(\Omega)$. In a problem I'm considering, it is stated that $|u_{j}|^{p}\nabla u_{j}\rightarrow |u|^{p}\nabla u$ in $L^{2}(\Omega)$, for certain $p\geq 1$, leaving it to the reader a size condition on $p$ for dimension $n\geq 1$ fixed. I don't know see why this should be true for general $n$.
If $n\leq 2$, then there is no issue by Sobolev embedding and Holder's inequality. Suppose $n\geq 3$. Observe that $$\nabla(|u_{j}|^{p+1})=(p+1)|u_{j}|^{p-1}u_{j}\nabla u_{j}$$
If $|u_{j}|^{p}\nabla u_{j}\in L^{2}(\Omega)$, then $\nabla(|u_{j}|^{p+1})\in L^{2}(\Omega)$, so by Poincare inequality $|u_{j}|^{p+1}\in L^{2}(\Omega)$. Suppose $p=1$, then the preceding implies that $u_{j}\in L^{4}(\Omega)$. I don't see why this should hold a priori, since if $n$ is very large then Sobolev embedding will not give that $u_{j}\in L^{4}(\Omega)$.
Any thoughts? Does anyone take issue with my argument above?