Product of two $W^{1,p}_0$ functions

Question

I have a fairly simple question. Suppose that $p>n$ and $u,v\in W^{1,p}_0$, how can I prove that the product is also in $W^{1,p}_0$ ? Of course, we have to employ Morrey's inequality. My idea was: $u$ and $v$ have uniformly continuous representatives. We can estimate the Hölder norm of the product in terms of the product of the Hölder norms. We can also use Hölder's inequality to ensure the product has finite $L_p$ norm. The only piece missing is that we have to show that the weak derivative exists. For this we need some kind of non-trivial product rule. And secondly, I am not really sure why Morrey's inequality should help us (this was hinted), since Morrey requires our function to be $W^{1,p}_0$ already. Anyway, thank you all :)