Let $1<p<\infty$, $\Omega\subset\mathbb{R}^m$ be open, bounded with $\partial\Omega\in C^1$. Assume that $u_k\to u$ weakly in $W^{1,p}(\Omega;\mathbb{R}^n)$. We know that $u_k\to u$ strongly in $W^{1,q}(\Omega;\mathbb{R}^n)$ for all $q$ such that $1<q<p<\infty$. Lets denote $u=(u^1,\ldots,u^n)$.
I want to prove the following convergence $$ \int_\Omega |\nabla u_k|^{p-2}\nabla u_k^i u_k^j \cdot \nabla v \, dx \to \int_\Omega |\nabla u|^{p-2}\nabla u^i u^j \cdot \nabla v \, dx $$ for all $v\in C_0^\infty(\Omega;\mathbb{R})$ and all $i,j=1,\ldots,n.$
It is supposed to be an easy task, but I struggle with it for some time. I have tried adding and substracting things under integral, but it becomes even messier to work with...