  In this paper we study a nonlinear magneto-static model on a general domain
which is multiply-connected and has $m$ holes, and under a nonlinear relation
between magnetic induction $\bold B$ and magnetic field $\bold H$. The equation
contains a Neumann field $\bold h_1\in\Bbb H_1(\Omega)$ and a Dirichlet field
$\bold h_2\in\Bbb H_2(\Omega)$, which represent the effects of domain topology.
For a general electric current, the equation contains an unknown gradient
$\nabla p(x)$, which represents the electric field. Existence results of
solutions of boundary value problems of this model under various types of
boundary conditions are proved, which exhibit the effects of domain topology.
