I have a function $u\in H^1_0(\Omega)$, $\partial\Omega$ being bounded and as nice as you need, and I need to prove that $u^k\in H^1_0(\Omega) $ for a fixed $k>1$ (integer if this can help).
Certainly, this requires additional assumptions on $u$. For instance, if I know that $u$ is continuous up to the boundary, then this is also true of $u^k$ and gives the necessary conclusion. But expressing this condition in the Sobolev scale requires to assume that $u$ belongs to $H^s$ for $s$ larger than $n/2$. Are there weaker additional conditions on $u$ which ensure that $u^k\in H^1_0(\Omega)$?