The following is stated in the paper

*Moshe Marcus and Victor J. Mizel*, MR 531975 **Complete characterization of functions which act, via superposition, on Sobolev spaces**, *Trans. Amer. Math. Soc.* **251** (1979), 187--218.

as Theorem 1 (note that $T_f$ is the superposition operator corresponding to $f$ and that the paper assumes $\Omega$ to satisfy the cone condition):

Suppose that $\Omega$ is bounded. Let $f \colon \mathbb R^m \to \mathbb R$ be a Borel function and let $p$, $r$ be two such that $1 \le r \le p < N$. Then $T_f$ maps $W^{1,p}(\Omega)^m$ into $W^{1,r}(\Omega)$ if and only if the following conditions hold:

- $f$ is locally Lipschitz in $\mathbb R^m$
- The first order partial derivatives of $f$ satisfy the inequality
$$ \left| \frac {\partial f}{\partial \xi_i}(\xi) \right| \le a_0(1+|\xi|^\nu)\ \text{a.e. in $\mathbb R^m$, $i = 1,\dotsc,m$}$$
where $a_0$ is a constant and $\nu = N(p-r)/(r(N-p))$.

If $N < p$ (or $N = 1$ and $1 \le p$) and $1 \le r \le p$ then $T_f$ maps $W^{1,p}(\Omega)^m$ into $W^{1,r}(\Omega)$ if and only if condition 1 holds.

The theorem goes on to state that under these circumstances $T_f$ is a continuous operator; bounds are established for its image, too. The case of unbounded $\Omega$ is considered in the same paper, see Theorem 3.