  We study the insulated conductivity problem with closely spaced insulators
embedded in a homogeneous matrix where the current-electric field relation is
the power law $J = |E|^{p-2}E$. The gradient of solutions may blow up as
$\varepsilon$, the distance between insulators, approaches to 0. In 2D, we
prove an upper bound of the gradient to be of order $\varepsilon^{-\alpha}$,
where $\alpha = 1/2$ when $p \in(1,3]$ and any $\alpha > 1/(p-1)$ when $p > 3$.
We provide examples to show that this exponent is almost optimal. In dimensions
$n \ge 3$, we prove an upper bound of order $\varepsilon^{-1/2 + \beta}$ for
some $\beta > 0$, and show that $\beta \nearrow 1/2$ as $n \to \infty$.
