# Weak convergence in Sobolev Space: does this integral converge to \$0\$?

Let $$Omega$$ be an open bounded domain in $$mathbb{R^N}$$ and $$A(x)in L^{infty}(Omega)$$. Let $$sgeq 1$$ and $$(u_n)_n$$ be a sequence such that
$$|u_n|^s u_nrightharpoonup|u|^s uquadmbox{ in } W_0^{1,p}(Omega).$$
If we suppose $$u=0$$, it is true that
$$int_{Omega} A(x) nabla(|u_n|^s u_n) dxto 0?$$

About me the answer is yes, I reasoned in this way. Since $$|u_n|^s u_nrightharpoonup|u|^s u$$ in $$W_0^{1, p}(Omega)$$ and $$u=0$$, thus $$|u_n|^s u_nrightharpoonup 0$$ in $$L^p(Omega)$$ and $$nabla(|u_n|^s u_n)rightharpoonup 0$$ in $$L^p(Omega)$$. Moreover, since $$1in L^p(Omega)$$ and since $$A(x)in L^{infty}(Omega)$$, a constant $$cgeq 0$$ exists such that
$$int_{Omega} |A(x)| nabla(|u_n|^s u_n) dxleq c int_{Omega} nabla(|u_n|^s u_n) dxto 0.$$