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A non-local thermistor problem. (English) Zbl 0826.35120
Summary: We consider the question of the existence/nonexistence of solutions for the nonlocal 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, $$L^{2, \mu}$$ estimates and fixed point arguments. Nonexistence is obtained by averaging procedures and an application of Barta’s Inequality.

##### MSC:
 35Q60 PDEs in connection with optics and electromagnetic theory 35J60 Nonlinear elliptic equations 80A20 Heat and mass transfer, heat flow (MSC2010)
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