Given the p-poisson equation ($1<p<\infty$): $$ -\Delta_p u=f \mbox{ in $B_r(0)\subset \mathbb R^N$} $$ with $f\in L^q(B_r)$ and $q>N$, I wish to show that:
$$ |\nabla u(0)|\leq C\left ( \frac{1}{r}\sup_{B_r}|u| + ||f||_{L^q(B_r)} \right ) $$.
Where $C$ is independent of $r$.
PS: I am talking about the equation (6.7) in M. Hayouni's Lipschitz Continuity of the state function in a shape Optimization Problem. Through Embedding Theorems, I could arrive up to uniform boundedness of gradient, but how to proceed for estimate (6.7). I expect it to be true for a general $p>1$
This is only a partial answer, or more of a suggestion. One standard trick to prove regularity for the $p$-Laplace equation is to regularize it in the following way
$$-\text{div} ( (|\nabla u|^2 + \epsilon^2)^{(p-2)/2}\nabla u) = f.$$
This operator is uniformly elliptic, and solutions are smooth. So you can differentiate the equation and try to use the maximum principle to bound the gradient. The idea is to look for estimates independent of $\epsilon>0$, since they hold for the $p$-Laplace equation as well. This should work to get gradient estimates, though they might have a bit of a different form from what you are asking.