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

我们研究了具有规定内部界面条件的非线性椭圆系统。这些模型经常用于物理系统中，其中离子转移起着重要作用，例如，在纳米层生长或锂电池建模中。该模型的主要困难在于在域内或界面上的本构方程中非线性的快速或非常缓慢的增长。在界面上，可以通过证明一种解决方案的最大原理来避免困难，但在域内，通量的这种规律性原则上不可用，因为本构律关于空间变量是不连续的。本文的主要结果是针对这些问题的存在性理论，在该理论中，我们要求引导函数满足德尔塔二条件或纳布拉二条件。