Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and $p\in (1,\infty)$. Let $-\Delta_p :W_0^{1,p}\to W^{-1,p'}$ be the $p$-Laplace operator, i.e. $$\langle-\Delta_pu,v\rangle=\int_\Omega |\nabla u|^{p-2}\nabla u\nabla v$$
where $\frac{1}{p}+\frac{1}{p'}=1$. Suppose that $f\in L^{p'}$, where. Assume that $u\in W_0^{1,p}$ satisfies \begin{equation}\tag{1}\langle-\Delta_p u,v\rangle=\int_\Omega fv,\ v\in W_0^{1,p}\end{equation}
Can I conclude that $u\in L^\infty$ ?
Remark 1: If $f\in L^{q}$ with $q>\frac{p}{N}$, then for $u\in W_0^{1,p}$ satisfying $(1)$ we have that $u\in L^\infty$ (see for example Serrin's paper: Local behavior of solutions of quasilinear equations - Theorem 3). But this theorem dont cover all cases.
Remark 2: I am asking this question, because in the paper of Loc and Schmitt - Applications of Sub-Supersolutions Theorems to Singular Nonlinear Elliptic Problems, they do this statament (Lemma 2.3). They ask the reader to look into the book of Ladyzhenskaya and Ural'tseva, but I was not able to find it there.
Remark 3: If $p>N$, then $u\in L^\infty$ automatically, so we can consider only the case $p\leq N$.
Thank you.
