How to prove the convergence of this integral?

Question

I rewrite here a question I made before because it had some mistakes before.

Let $p > 2$ and $\Omega \subset R^n$ an open bounded subset. Let $(u_n)_n, v\in W_0^{1, p}(\Omega)$ such that \begin{align*} \int_{\Omega} \vert\nabla u_n \vert^p dx \rightarrow 0 \quad \mbox{ and } \quad \Vert v\Vert_{W_0^{1,p}}\leq 1. \end{align*} I want to prove that \begin{align*} \int_{\Omega} \vert\nabla u_n\vert^{p - 2} \vert \nabla u_n\cdot\nabla v\vert dx \rightarrow 0. \end{align*} I am proceeding in this way (by using Cauchy - Schwartz inequality), but I'm not sure that this is right: \begin{align*} \int_{\Omega} \vert\nabla u_n\vert^{p - 2} \vert \nabla u_n\cdot\nabla v\vert dx \leq \int_{\Omega} \vert\nabla u_n\cdot\nabla v\vert^p dx \left(\int_{\Omega}\vert\nabla u_n\vert^p dx\right)^{\frac{p - 2}{p}} \rightarrow 0. \end{align*} Could anyone help?

Answer 1

Yes, this time the result is true. I think you should rather write $$ \int_\Omega |\nabla u_n|^{p-2} |\nabla u_n\cdot\nabla v| \leq \int_\Omega |\nabla u_n|^{p-1} |\nabla v|. $$ And now you can use Hölder's inequality to get $$ \int_\Omega |\nabla u_n|^{p-1} |\nabla v| \leq \|\nabla u_n\|_{L^p}^{p-1} \|\nabla v\|_{L^p}, $$ which goes to zero since $\|\nabla v\|_{L^p}$ is a finite constant and $\|\nabla u_n\|_{L^p} \to 0$.