Weakly convergence in $W^{1,p}_0$ and strong convergence in $L^p$

Question

I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ that converges weakly to $u\in W^{1,p}_0(\Omega)$ and converges strongly to $u$ in $L^p(\Omega)$. We define a function $f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ a bounded Carathéodory 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 use the Cauchy-Schwarz inequality, but when $p$ is not equal to 2, I have no idea.

Thank you

Answer 1

Let $K$ be the upper bound for $f$, and assuming $\Omega$ is a bounded domain, and $1\le p\le\infty$, we have in particular that $u_n-u\in L^1(\Omega)$, so $$|\int_\Omega f(x,u_n)(u_n-u)\,dx|\le K\|u-u_n\|_{1,\Omega}\to 0$$ as $n\to\infty$.

Also $$\int_\Omega |u_n|^{p-2}u_n(u_n-u)\,dx\le\int_\Omega|u_n|^{p-1}|u_n-u|\,dx$$ $$\le\|u_n\|_{p,\Omega}^{p-1}\|u_n-u\|_{p,\Omega}\to 0$$ as $n\to\infty$