  In the last paper \cite{R7}, it was studied Hilbert, Poincare and Neumann
boundary-value problems with arbitrary measurable data for generalized analytic
functions and generalized harmonic functions with applications to the relevant
problems of mathematical physics. The present paper is devoted to the study of
the boundary-value problems with arbitrary measurable boundary data in domains
with rectifiable boundaries for the corresponding semi-linear equations with
suitable nonlinear sources. For this purpose, here it is constructed completely
continuous operators generating nonclassical solutions of the Hilbert and
Poincare boundary-value problems with arbitrary measurable data for the Vekua
type equations and the Poisson equations, respectively. On this base, it is
first proved the existence of solutions of the Hilbert boundary-value problem
with arbitrary measurable data in any domains with rectifiable boundaries for
the nonlinear equations of the Vekua type. It is necessary to note that our
approach is based on the geometric interpretation of boundary values as angular
(along nontangential path) limits in comparison with the classical variational
approach in PDE. The latter makes it is also possible to obtain the theorem on
the existence of nonclassical solutions of the Poincare boundary-value problem
on the directional derivatives and, in particular, of the Neumann problem with
arbitrary measurable data to the Poisson equations with nonlinear sources in
Jordan domains with rectifiable boundaries. As consequences, then it is given a
series of applications of this theorem to some problems of mathematical physics
describing, for instance, such phenomena as physical and chemical absorption
with diffusion, plasma states, stationary burning etc.
