# zbMATH — the first resource for mathematics

Existence and regularity of solutions of a nonlinear nonuniformly elliptic system arising from a thermistor problem. (English) Zbl 0830.35045
The author studies the steady state thermistor problem, which consists of an elliptic system $\nabla (\sigma (u) \nabla \varphi) = 0,\quad (2) - \nabla (k(u) \nabla u) = \sigma (u) |\nabla \varphi |^2,\quad \text{in } \Omega,\tag{1}$ and boundary conditions (3) $$u = \varphi_0$$ on $$\Gamma^\varphi_D$$, $$\partial_n \varphi = 0$$ on $$\Gamma^\varphi_N \equiv \partial \Omega \backslash \overline {\Gamma^\varphi_D}$$, (4) $$u = u_0$$ on $$\Gamma^u_D$$, $$\partial_nu + h(x,u) = 0$$ on $$\Gamma^u_N \equiv \partial \Omega \backslash \overline {\Gamma^u_D}$$, where $$\Omega$$ is a domain in $$\mathbb{R}^N$$, $$\partial_n$$ is the outward normal derivative and $$\Gamma^\varphi_D$$, $$\Gamma^u_D$$ are smooth surfaces. This problem has many applications such as current regulation, switching, thermal conductivity analysis, etc. In the paper, $$\sigma(u)$$ is not necessarily uniformly bounded from above and away from zero as $$u \to \infty$$, (this causes the main difficulty to study (1)–(4), together with the quadratic growth in (2)). Under certain additional restrictions on $$\sigma (u)$$ and $$h(x,u)$$, it is shown the existence and Hölder continuity of the solution. The proofs use the classical Nash-Moser iteration (to establish $$L^\infty$$ bounds) and some results from a method exposed by M. Giaquinta [Multiple integrals in the calculus of variations and nonlinear elliptic systems (Princeton, New Jersey) (1983; Zbl 0516.49003)]. The paper finishes with a non-existence and nonuniqueness example.