In this paper an existence theorem of positive solutions to the Dirichlet problem for elliptic equations having nonlinear terms with an uncountable set of discontinuities is established. Some applications to special cases, such as problems with critical Sobolev growth, are also presented. The approach taken is strictly based on set-valued analysis and fixed point arguments.

We prove that a variational quasilinear elliptic equation admits a positive weak solution on $\mathbb R^n$. Our results extend to a wider class of equations some known results about semilinear and quasilinear problems: all the coefficients involved (also the ones in the principal part) depend both on the variable $x$ and on the unknown function $u$; moreover, they are not homogeneous with respect to $u$.

The method of a priori estimates based on the Leray–Schauder topological degree theory is developed to establish the existence of solutions to general boundary value problems for differential equations with "maxima".

The paper [Topol. Methods Nonlinear Anal. 4 (1994), 79–136] from the title was very long and rather technical. Nevertheless it has drawn attention of some people who have pointed out to the author some mistakes and inaccuracies. Especially, the author thanks A. P. Sadovski for sending him a list of mistakes. Also the author himself found some misprints, mistakes and one incomplete proof. Moreover, some new results about this subject have appeared meantime.