The insulated conductivity problem with $p$-Laplacian

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 $J=\mathrm{\mid }E{\mathrm{\mid }}^{p-2}E$. The gradient of solutions may blow up as $\epsilon$, the distance between insulators, approaches to 0. In 2D, we prove an upper bound of the gradient to be of order ${\epsilon }^{-\alpha }$, where $\alpha =1\mathrm{/}2$ when $p\in (1,3]$ and any $\alpha >1\mathrm{/}(p-1)$ 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 ${\epsilon }^{-1\mathrm{/}2+\beta }$ for some $\beta >0$, and show that $\beta \nearrow 1\mathrm{/}2$ as $n\to \mathrm{\infty }$.