Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e $\Omega$. How we can use the Fatou lemma to prove that $f_n(x,u_n)u_n$ strongly converges in $L^2$ to $f(x,u)u$ and a.e in $\Omega$?
Thanks for the help.