Let $\Omega\subset \mathbb{R}^2$ be a bounded smooth domain. Let $u\in W^{k,p}(\Omega)$ be a non-negative $k-$times weakly differentiable function where $p\geq 1$ is arbitrary. Now let $q>0$ be a positive number. My question is that does $u^q\in W^{k,p}(\Omega)$ too?
In fact, I'm dealing with the following problem:
Let $u\in H_0^1(\Omega)$ be a non-negative solution of the following equation:
$$ -\Delta u=\lambda u^{q}, $$ where $\lambda,q>0$. I want to prove that $u\in C^\infty$.
$\textbf{My attempt:}$ Since we are in dimension $2$ and $\Omega$ is smooth, it's natural to know that $u\in C^1$ by the $W^{2,p}$ estimate and Sobolev embedding. Then I'm stuck since I don't know how to prove the infinity smoothness. So if the above proposition is right, things will become more trivial. But if it fails, how can I get the smoothness of $u$?
Any hints and help will be appreciated a lot!