After passing to a subsequence, weak compactness and the <Rellich-Kondrachov compactness theorem> give
$$
u_{n_k}\rightharpoonup u\quad\hbox{in }H^1(U),
\qquad
u_{n_k}\longrightarrow u\quad\hbox{in }L^3(U).
$$
For fixed $w\in H^1(U)$, the <Sobolev inequality> gives $w\in L^6(U)$, so
$$
u_{n_k}w\longrightarrow uw\quad\hbox{in }L^2(U).
$$
Meanwhile $\partial_j u_{n_k}\rightharpoonup\partial_j u$ in $L^2(U)$. Pairing this weak convergence with the strong convergence of the products, or equivalently using the <weak-strong product convergence lemma>, yields
$$
\int_U(\partial_xu_{n_k}+\partial_yu_{n_k}+\partial_zu_{n_k})u_{n_k}w
\longrightarrow
\int_U(\partial_xu+\partial_yu+\partial_zu)uw.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 105 / 2 / b / iii / Solution

If $f_n\rightharpoonup f$ in $L^p$, $g_n\to g$ in $L^q$, and $1/p+1/q=1$, then
$$
\int f_ng_n\longrightarrow\int fg.
$$
Indeed, split the difference into the weak pairing with $g$ and a term bounded by the <Holder inequality> using $g_n-g$.

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 Incoming links: Weak-strong product convergence lemma