Let $\Omega\subset \mathbb{R}^n$ be a bounded domain, $p\in (1,\infty)$. Suppose that $u_n\in L^p(\Omega)$. By using the fact that $L^p(\Omega)$ is uniformly convex, we know that if $u_n\rightharpoonup u$ and $\|u_n\|_p\rightarrow \|u\|_p$, then $u_n\rightarrow u$.
Now, if $u_n\rightharpoonup u$ and $\|\sqrt{f^2+u_n^2}\|_p\rightarrow\|\sqrt{f^2+u^2}\|_p$ with $f\in L^p(\Omega)$, can we conclude the same thing, i.e. $u_n\rightarrow u$?
Thanks