**Note from the answerer** : this question stems from this article.

I ask this question in https://math.stackexchange.com/questions/1206617

I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so it weakly converge to $u\in W^{1,p}_0(\Omega)$ and strongly converge to $u$ in $L^p(\Omega).$ We define a function $f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ a bounded Caratheodory function such that $\lim_{s\rightarrow+\infty} f(x,s)=f^{+\infty}(x)$

My question is why $$\lim_{n\rightarrow +\infty} \int_{\Omega}f(x,u_n)(u_n-u) dx=0$$ and $$\lim_{n\rightarrow +\infty}\int_{\Omega} |u_n|^{p-2} u_n(u_n-u) dx=0$$

For the first integral, I'm trying to apply Lebesgue dominated convergence, but I have no idea.

For the second integral, when $p=2$ I have no problems, because in this case we have not $|u_n|^{p-2}$ it is equal to 1 and then I just have to do $u_n(u_n-u)=(u_n-u+u)(u_n-u)$ and I do the Cauchy–Schwarz inequality, but when $p$ is not equal to 2 I have no idea.

Thank you