Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. Is it true that if $(\varphi_n)_{n\geq 1}\subset C^{\infty}_c(\Omega)$ with $\varphi_n\to \varphi$ in $W^{1,p}(\Omega)$ and $\varphi\in W^{1,p}_0(\Omega)$ then we have that:

$$\lim\limits_{n\to\infty} \int_{\Omega} u(x)\varphi_n(x)\ dx=\int_{\Omega} u(x)\varphi(x)\ dx$$

for any $u\in W^{1,p}_0(\Omega)$?

I cannot find a counterexample. I think that this is not true probably for any value of $p$ but if $p$ satisfies some condition, like $p\geq \dfrac{2d}{d+1}$ it will hold maybe...

I posted this question on MSE at this link.