If $u\in W^{1,p}(\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ and $\xi$ is a smooth compactly supported function in $\Omega$, is it true that $\xi u^{\beta-p+1} \in W^{1,p}_0$ if $\beta >p-1$? (In the end my problem is to say if $u^{\beta-p+1} \in W^{1,p}$ (from this I know it follows the result).)
I think if $\Omega$ is not bounded we can't say, but if it is bounded, then we know the function $u$ belongs to $L^r$ for $r , but $\beta>p-1$ could be also greater than p. Maybe if I add the hypotesys $u\in L^{\infty}$ I could conclude? Thanks for any help.