# Weak Convergence from Strong Convergence

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 $$2. I want to show that $$\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!

• Since your domain is one-dimensional, the convergence in $H_0^1(\Omega)$ implies convergence in $L^\infty(\Omega)$. – gerw Mar 20 at 6:33
• I want to ensure. Do you mean I can take $\sup_{\Omega}|u_{n}|u_{n}|^{p-2} - v|v|^{p-2}|\to 0$ as $n\to\infty$? How do I show this claim? Thank you for your hint. – Evan William Chandra Mar 25 at 4:18
• This should follow from $\sup_\Omega | u_n - v| \to 0$ (together with the uniform boundedness of all involved functions). – gerw Mar 25 at 7:02