Let $n\ge 2$ be a natural number and $p<n$. Suppose $f\in W_0^{1,p}(\Omega)$ where $\Omega\subseteq \mathbb{R}^n$ is a bounded domain.
Then, by the Sobolev inequality, $$ ||f||_{p^\ast} = ||f||_{\frac{np}{n-p}} \le C||{Df}||_{p} $$ where $||\cdot||_q$ refers to the $L^q$-norm on $\Omega$.
Does the Sobolev inequality remain valid for functions $|f|^\alpha$ when $f\in W^{1,p}_0(\Omega)$ and $\alpha \ge 1$. I.e. if $f\in W^{1,p}_0(\Omega)$ can I conclude that $|||f|^\alpha||_{p^\ast}\le C|| D(|f|^\alpha)||_p$? In particular, if the left hand side is infinite then so is the right-hand side.
I was thinking we can define $G_N : [0, \infty)\to [0, \infty)$ to be a $C^1$ function such that $G_N(x) = x^\alpha$ for $x \le N$, and $G_N$ is linear on $[N, \infty)$.
Then, given that $f \in W_0^{1,p}(\Omega)$ we also have $|f| \in W_0^{1,p}(\Omega)$ and, since $G_N$ is $C^1$ with bounded derivtive and $G_N(0) = 0$, $G_N(|f|)\in W_0^{1,p}(\Omega)$. Ergo, $$ ||G_N(|f|)||_{p^\ast} \le C||D[G_N(|f|)]||_{p}. $$ Letting $N\to\infty$ on either side (using the monotone convergence theorem), I should obtain $$ |||f|^\alpha||_{p^\ast}\le C|| D(|f|^\alpha)||_p $$