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We study a nonlinear elliptic system with prescribed inner interfaceconditions. These models are frequently used in physical system where the iontransfer plays the important role for example in modelling of nano-layer growthor Li-on batteries. The key difficulty of the model consists of the rapid orvery slow growth of nonlinearity in the constitutive equation inside the domainor on the interface. While on the interface, one can avoid the difficulty byproving a kind of maximum principle of a solution, inside the domain suchregularity for the flux is not available in principle since the constitutivelaw is discontinuous with respect to the spatial variable. The key result ofthe paper is the existence theory for these problems, where we require that theleading functional satisfies either the delta-two or the nabla-two condition.This assumption is applicable in case of fast (exponential) growth as well asin the case of very slow (logarithmically superlinear) growth.

Existence and qualitative theory for nonlinear elliptic systems with a nonlinear interface condition used in electrochemistry