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 anddiscontinuous 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 solutionsfrom 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 conditionfor semiregularity of generalized solutions of this problem is given.The constraints on the discontinuous nonlinearityare relaxed in comparison with those used by H. J. Kuiper and K. C. Chang.

quasilinear elliptic type equation, parameter, discontinuous nonlinearity, continuum of positive solutions, semiregular solution, topological method