Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain and $f\in L^q$ with $q\in (1,\infty)$. If $u\in H_0^1(\Omega)$ satisfies $\int_\Omega \nabla u\nabla v=\int_\Omega fv,\ \forall\ v\in H_0^1(\Omega)$
then we can conclude that there exist some constant $C>0$ such that $\|u\|_{1,q}\leq C(\|u\|_q+\|f\|_q)$
Now suppose that $p\in (1,\infty)$ and $u\in W_0^{1,p}(\Omega)$ satisfies $\int_\Omega |\nabla u|^{p-2}\nabla u\nabla v=\int_\Omega fv,\ \forall\ v\in W_0^{1,p}(\Omega)$
Can we conclude that there exist some constant $C>0$ such that $\|u\|_{1,q}\leq C(\|u\|_q+\|f\|_q)$
Notes: The constant $C$ does not depend on $q$ and $u$. For the case $p=2$, we can show that in fact $u$ is a Strong solution i.e. $-\Delta u=f$ almost everywhere. From this fact we can conclude the first inequality (see for example Gilbard-Trudinger in the section of $L^p(\Omega)$ estimates Chapter 9).
For $p\neq 2$ I dont know if $u$ still being a Strong solution and if the same techinique can be applied.
Any hint or reference would be appreciated.