A non-local thermistor problem

Abstract

We consider the question of the existence/non-existence of solutions for the non-local nonlinear elliptic system which models a thermistor driven by a current source. Specifically, we show that for small input current there exists a solution, while this will not in general be the case for a sufficiently large current. A feature of our estimates is that the conditions for non-existence are determined by local criteria on the domain and the coefficients. Our basic tools for existence involve truncation, L2.μ estimates and fixed point arguments. Non-existence is obtained by averaging procedures and an application of Barta's Inequality.