Existence results for a nonlinear elliptic system modeling a temperature dependent electrical resistor (Q1122058)

The authors study the particular nonlinear elliptic system: \[ \nabla \cdot (\sigma (u)\nabla \phi)=0,\quad -\nabla (k \nabla u)=\sigma (u)| \nabla \phi |^ 2, \] for the temperature u and electric potential \(\phi\), modelling the steady state behaviour f a thermally sensitive resistor (\(\sigma\), k denote electrical and thermal conductivity). The boundary values of u, \(\phi\) are held constant on the smooth boundary of the body under consideration. We note that the system displays a quadratic dependence on the electric field \(\nabla \phi\). Based on the Leray-Schauder fixpoint principle and an approximation procedure existence of a weak solution is obtained.