On the Structure and Stability of the Set of Solutions of a Nonlocal Problem Modeling Ohmic Heating

In this work we analyze the existence, structure and stability of the set of positive solutions of a nonlocal elliptic boundary value problem modeling Ohmic heating. By regarding the total electric current flowing through the device by unit of cross-sectional area as the continuation parameter, we show that the set of steady states is constituted by a finite number of differentiable curves. Moreover, the instability index of a steady state changes by 1 when we pass through by a turning point and does not change if we pass through an hysteresis point. Some sufficient conditions so that the model admit a positive steady state for each value of the parameter, as well as for uniqueness and non-existence, will be also given.