First consider the following two Sobolev Embedding Theorems.
Theorem 1:
The continuous embedding $W^{1,p}(\Omega) \subset L^{p^{*}}(\Omega)$ holds provided the exponent $p^{*}$ is defined as $p^{*} := \{ \frac{np}{n-p} \text{ for } p <n, \text{ an arbitrary large real for } p=n, + \infty \text { for } p>n \}$
Theorem 2(Rellich, Kondrachov Theorem):
The compact embedding $W^{1,p}(\Omega) \Subset L^{p^{*}-\epsilon}(\Omega)$ for $\epsilon \in (0,p^{*}-1]$ holds for $p^{*}$.
Consider a bounded continuous mapping $f$:
$f: L^{p^{*}-\epsilon}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow L^{p'}(\Omega;\mathbb{R}^{n})$ $\text{ where } p^{'}=\frac{1}{p-1}$
How does it follow then that we can state that: $f: W^{1,p}(\Omega)\times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow L^{p'}(\Omega;\mathbb{R}^{n})$ is (weak $\times$ norm, norm) -continuous?
Thanks for assistance. Let me know if something is unclear.
Assume that $w_\alpha=(x_\alpha,y_\alpha)$ is a net in $W^{1,p}(\Omega)\times L^p(\Omega:\mathbb{R}^n)$ such that $w_\alpha\to w=(x,y)$. We need to prove that $$f(w_\alpha)\to f(w)\tag{1}$$
We have that $x_\alpha\to x$ in the weak topology of $W^{1,p}(\Omega)$. Once Theorem 2 is valid, we can conclude that $x_\alpha \to x$ in the strong topology of $L^{p^\star-\epsilon}(\Omega)$, hence, as $f$ is continuoous with respesct to norm$\times$norm topology, we conclude that $(1)$ is true.
Remark: there is a assumption in my proof which must be verified.