Local boundedness for $p$-Laplacian with degenerate coefficients

We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by $\mathrm{\nabla }\cdot (\lambda |\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)=0$, where the variable coefficient $0\le \lambda$ and its inverse ${\lambda }^{-1}$ are allowed to be unbounded. Assuming certain integrability conditions on $\lambda$ and ${\lambda }^{-1}$ depending on $p$ and the dimension, we show local boundedness. Moreover, we provide counterexamples to regularity showing that the integrability conditions are optimal for every $p>1$.