Let $U \subset \mathbb{R}^N$ be a bounded open set and let $\{ U_j \}_{j=1}^\infty$ be an open covering of $U$ such that $U = \bigcup\limits_{j=1}^\infty U_j$.
Suppose $\{ \psi_j \}_{j=1}^\infty$ is a partition of unity subordinate to $\{ U_j \}_{j=1}^\infty$.
Let $u \in W^{1,p}(U)$ and let $h \in C_c^\infty(\mathbb{R}^N)$ be such that $\mathrm{supp} \, h \cap U \neq \emptyset$.
In $U$ we have $h = \sum\limits_{j=1}^\infty \psi_j h$ — since $\sum\limits_{j=1}^\infty \psi_j = 1$ in $U$.
I want to prove the following identity (if it holds): $$\lim_{n \rightarrow \infty} \int_{U} \vert \nabla u \vert^{p-2}\nabla u \cdot \nabla \left( \sum_{j=1}^n \psi_j h \right) = \int_{U} \vert \nabla u \vert^{p-2}\nabla u \cdot \nabla h.$$
I failed using the dominated convergence theorem: the main difficulty is to find $g \in L^1(U)$ such that — almost everywhere on $U$ and for every $n$ — there holds $$\vert \nabla u \vert^{p-1} \bigg\lvert \sum_{j=1}^n \nabla (\psi_j h) \bigg\rvert \leq g.$$
I'm not sure the above identity is valid; what I'm trying to do is replace $h$ whose support is not necessary entirely in $U$ by a function $\widetilde{h} \in C_c^\infty(U)$ such that, assuming $u = 0$ outside $U$, we have $$\int_{\mathrm{supp} \, h \cap U} \vert \nabla u \vert^{p-2}\nabla u \cdot \nabla h = \int_U \vert \nabla u \vert^{p-2}\nabla u \cdot \nabla \widetilde{h}.$$