| Abstract: | We shall give the existence of a capacity solution to a nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $$varphi $$, the model problem we refer to is $$begin{aligned} left{ begin{array}{l} Delta _p u+g(x,u)= rho (u)|nabla varphi |^2 quad mathrm{in} quad Omega , {{,mathrm{div},}}(rho (u)nabla varphi ) =0 quad mathrm{in} quad Omega , varphi =varphi _0 quad text{ on } quad {partial Omega }, u=0 quad mathrm{on} quad {partial Omega }, end{array} right. end{aligned}$$where $$Omega subset mathbb {R}^N$$, $$Nge 2$$ and $$Delta _p u=-{text {div}}left( |nabla u|^{p-2} nabla uright) $$ is the so-called p-Laplacian operator, and g a nonlinearity which satisfies the sign condition but without any restriction on its growth. This problem may be regarded as a generalization of the so-called thermistor problem, where we consider the case of the elliptic equation is non-uniformly elliptic. |
