Existence of capacity solution for a perturbed nonlinear coupled system

The aim of this paper is to show the existence of a capacity solution to a degenerate perturbed system involving an equation of parabolic type and an equation of elliptic type. This system may be regarded as a generalization version of the well-known thermistor problem; in this case, the unknowns are the temperature in a conductor and the electrical potential. We study the general case where the nonlinear elliptic operator in the parabolic equation is of the form Au=-div(a(x,t,∇u)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Au =- \mbox{div}( a(x,t,\nabla u)),$$\end{document}A being a Leray–Lions operator, which includes the particular case of the p-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p-$$\end{document}Laplacian operator. The problem is also perturbed by a function satisfying a sign condition and without assuming any restriction on its growth.