The Nemytskii mappings in Lebesgue spaces theorem is as follows:
If $a: \Omega \times \mathbb{R}^{m_{1}} \times \cdots \times\mathbb{R}^{m_{j}} \rightarrow \mathbb{R}^{m_{0}}$ is a Caratheodory mapping and the functions $u_{i}: \Omega \rightarrow \mathbb{R}^{m_{i}}$ where $i=1,\ldots,j$ are measurable then $\eta_{a}(u_{1},\ldots,u_{j})$ is measurable. Moreover, if $a$ satisfies the growth condition: $|a(x,r_{1},\ldots,r_{j})| \leq \gamma(x) + \sum_{i=1}^{j}|r_{i}|^{\frac{p_{i}}{p_{0}}}$ for some $\gamma \in L^{p_{0}}(\Omega)$. With $1 \leq p_{i} < +\infty$, $1 \leq p_{o} < +\infty$, then $\eta_{a}$ is a bounded continuous mapping $L^{p_{1}}(\Omega;\mathbb{R}^{m_{1}}) \times \cdots \times L^{p_{j}}(\Omega;\mathbb{R}^{m_{j}}) \rightarrow L^{p_{0}}(\Omega; \mathbb{R}^{m_{0}})$.
This theorem produces an operator which is continuous. The continuity is with reference to the norm topology $\times \cdots \times$ norm topology $\rightarrow$ norm topology. If we consider the Sobolev spaces instead of the $L^{p}$ spaces, how could we use this theorem to get a continuous operator $W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{q}(\Omega)$?
Definition: Considering integers $j,m_{0},m_{1},...,m_{j}$ we say that a mapping $a: \Omega \times \mathbb{R}^{m_{1}} \times ... \times \mathbb{R}^{m_{j}} \rightarrow \mathbb{R}^{m_{0}}$ is a Caratheodory mapping if $a(\frac{}{},r_{1},...,r_{j}): \Omega \rightarrow \mathbb{R}^{m_{0}} $ is measurable for all $(r_{1},...,r_{j}) \in \mathbb{R}^{m_{1}} \times ... \times \mathbb{R}^{m_{j}}$ and $a(x,\frac{}{}): \mathbb{R}^{m_{1}} \times ... \times \mathbb{R}^{m_{j}} \rightarrow \mathbb{R}^{m_{0}}$ is continuous for a.a. $x \in \Omega$. Then the so-called Nemytskii mappings $\eta_{a}$ map functions $u_{i}: \Omega \rightarrow \mathbb{R}^{m_{i}}$, $i=1,...,j$ to a function $\eta_{a}(u_{1},...,u_{j}) : \Omega \rightarrow \mathbb{R}^{m_{0}}$ defined by $[\eta_{a}(u_{1},...,u_{j})](x) = a(x,u_{1}(x),...,u_{j}(x))$.
If you only need the continuity into $L^q$, then (I bet) the best thing you can do is to use the Sobolev embedding $W^{1,p} \hookrightarrow L^r$ and the continuity of $\eta_a : L^r \times L^p \to L^q$.