On certain doubly non-uniformly and singular non-uniformly elliptic systems Original Research Article

We consider the steady state of the thermistor problem consisting of a coupled set of nonlinear elliptic equations governing the temperature and the electric potential. We study the existence of weak solutions under two kind of assumptions. The first one considers the case in which the two diffusion coefficients are not bounded below far from zero, arising to a doubly non-uniformly elliptic system. In the second one, we assume in addition that the thermal conductivity blows up for a finite value of the temperature, arising to a singular and non-uniformly coupled system.