The existence and uniqueness properties for extremal mappings with smallest weighted Lp distortion between annuli and the related Grotzsch type problems are discussed. An interesting critical phase type phenomena is observed. When p < 1, apart from the identity map, minimizers never exist. When p = 1 we observe Nitsche type phenomena; minimisers exist within a range of conformal moduli determined by properties of the weight function and not otherwise. When p > 1 minimisers always exist. Interpreting the weight function as a density or "thickness profi le" leads to interesting models for the deformation of highly elastic bodies and tearing type phenomena.