I work on a bounded domain in $\mathbb{R}^n$ and let $p \geq 2$ and the operator $\Delta_p u = \nabla \cdot (|\nabla u |^{p-2}\nabla u)$.

Does the following inequality (or something similar hold) for $u \in W^{2,p}\cap W^{1,p}_0$: $$\lVert u \rVert_{W^{2,p}} \leq C(\lVert \Delta_p u \rVert_{L^2})$$ (with possibly a $\lVert u \rVert_{L^s}$ contribution on the RHS). So essentially I am looking for a elliptic regularity result.

I tried looking for elliptic regularity results regarding the $p-$laplace equation but had no good luck.

Consider for example $p=2$. Then $\lVert u \rVert_{H^2}$ is bounded above by a factor of $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$. In fact they are equivalent norms in $H^2$.