Let $\Omega \subset \mathbb{R}$ be a bounded domain, $v \in H_{0}^{1}(\Omega)$ such that $||u_{n}-v||_{H_{0}^{1}(\Omega)}\to 0$ as $n\to\infty$ for a bounded sequence $\{u_{n}\}_{n\in\mathbb{N}} \subset H_{0}^{1}(\Omega)$, and $2<p<\infty$. I want to show that $\forall \phi \in C_{0}^{\infty}(\Omega),\, \int_{\Omega}u_{n}|u_{n}|^{p-2}\phi dx \to \int_{\Omega}v|v|^{p-2}\phi dx$
This is my attempt so far $$|\int_{\Omega}(u_{n}|u_{n}|^{p-2} - v|v|^{p-2})\phi dx| \leq\sup\limits_{\Omega}|\phi|\int_{\Omega}|u_{n}|u_{n}|^{p-2} - v|v|^{p-2} |dx $$ Then, I am not sure how to proceed from that line to make use of the bounded sequence. I was only given hints to use mean value theorem and Holder inequality (which I have used by taking the supremum for $\phi$).
Any help is much appreciated! Thank you very much!