We derive closed-form expressions for the effective conductivity of a class of powerlaw “differential” laminates in arbitrary dimension, studied in previous works (Idiart et al, 2013). Emphasis is put on the weakly and strongly nonlinear regimes, and on the asymptotic behavior of such composite in the “dilute limit” of a vanishingly small volume fraction of one of the two phases. In the strongly nonlinear limit, we examine the behavior of these solutions as the partial derivative equations loose ellipticity. The domain of validity of the resulting expressions are examined, and we provide a geometric interpretation of the results, relevant to field localization. The behavior of the solutions at increasing dimensions, and whether they could represent bounds is examined.