About a problem on conductor heating

Kuiper's problem on conductor heating in a uniform electric field of intensity $\sqrt\lambda$ with a positive parameter $\lambda$ is considered. The conductor temperature distribution is a solution to the Dirichlet problem with homogeneous initial data in a bounded domain for a quasilinear elliptic equation with a discontinuous nonlinearity and a parameter. The heat-conductivity coefficient depends on the spatial variable and temperature, and the specific electric conductivity has discontinuities with respect to the phase variable. The existence of a continuum of generalized positive solutions that connects $(0,0)$ to $\infty$ is proved by a topological method. A sufficient condition is obtained for such solutions to be semiregular. Compared to the papers of H.J. Kuiper and K.C. Chang, the restrictions on the discontinuous nonlinearity (the specific electric conductivity) are weaken.