Given that $\Omega$ is bounded and $a_{ij}(u_{k}) \rightarrow a_{ij}(u)$ in $L^{1}(\Omega)$, $a_{i0}(u_{k}) \rightarrow a_{i0}(u)$ in $L^{1}(\Omega)$, $\frac{\partial u_{k}}{\partial x_{j}} \rightarrow \frac{\partial u}{\partial x_{j}} \text{ in } L^{1}(\Omega)$, $c_{j}(u_{k}) \rightarrow c_{j}(u) \text{ in } L^{1}$ and $c_{o}(u_{k}) \rightarrow c_{o}(u)$ in $L^{1}(\Omega)$ for all $i,j =1,...,n$. We also have $v$ and $\frac{\partial v}{\partial x_{i}}$ in $L^{p}(\Omega)$. How does it follow that the limit of the integral is the integral of the limit in the attached image below. I might be missing something simple.
Let me know if any additional info is required.