## Stationary solutions to the thermistor problem.(English)Zbl 0787.35033

The authors consider the nonlinear problem $-\nabla \cdot(k(u)\nabla u)=\sigma(u) | \nabla \varphi |^ 2,\quad \nabla \cdot(\sigma(u)\nabla \varphi)= 0 \tag{*}$ in a bounded domain $$\Omega \subset \mathbb{R}^ n$$, $$n \geq 2$$, with Lipschitz boundary and with general mixed boundary conditions, which is a mathematical model for the description of the steady state distribution of the temperature $$u$$ and the electrical potential $$\varphi$$ in an electric device (thermistor) whose electrical properties are temperature dependent.
The existence of a weak solution of $$(*)$$ for general boundary conditions is proved under the sole assumption that $$\sigma$$ is continuous and uniformly positive. Uniqueness of the solution is also proved for sufficiently small data. Then, a new nonlocal condition which relates to the way the device is connected to the rest of the electrical circuit is analyzed, and sufficient conditions for the existence of a weak solution for such a problem are derived. Finally, a sufficient condition for non- uniqueness of the problem in one space dimension is given.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35Q80 Applications of PDE in areas other than physics (MSC2000)
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