One class of quasilinear elliptic type equations with discontinuous nonlinearities

In a bounded domain $\Omega\subset \mathbb{R}^n$, a class of quasilinear elliptic type boundary problems with parameter and discontinuous nonlinearity is studied. This class of problems includes the H. J. Kuiper conductor heating problem in a homogeneous electric field. The topological method is applied to verify the existence of a continuum of generalized positive solutions from the Sobolev space $W_p^2(\Omega)$ ($p>n$) connecting $(0,0)$ with $\infty$ in the space $\mathbb R\times C^{1,\alpha}(\overline\Omega)$, $\alpha\in (0,(p-n)/p)$. A sufficient condition for semiregularity of generalized solutions of this problem is given. The constraints on the discontinuous nonlinearity are relaxed in comparison with those used by H. J. Kuiper and K. C. Chang.