  We study the Dirichlet problem for the semi--linear partial differential
equations ${\rm div}\,(A\nabla u)=f(u)$ in simply connected domains $D$ of the
complex plane $\mathbb C$ with continuous boundary data. We prove the existence
of the weak solutions $u$ in the class $C\cap W^{1,2}_{\rm loc}(D)$ if a Jordan
domain $D$ satisfies the quasihyperbolic boundary condition by Gehring--Martio.
An example of such a domain that fails to satisfy the standard (A)--condition
by Ladyzhenskaya--Ural'tseva and the known outer cone condition is given. We
also extend our results to simply connected non-Jordan domains formulated in
terms of the prime ends by Caratheodory. Our approach is based on the theory of
the logarithmic potential, singular integrals, the Leray--Schauder technique
and a factorization theorem in \cite{GNR2017}. This theorem allows us to
represent $u$ in the form $u=U\circ\omega,$ where $\omega(z)$ stands for a
quasiconformal mapping of $D$ onto the unit disk ${\mathbb D}$, generated by
the measurable matrix function $A(z),$ and $U$ is a solution of the
corresponding quasilinear Poisson equation in the unit disk ${\mathbb D}$. In
the end, we give some applications of these results to various processes of
diffusion and absorption in anisotropic and inhomogeneous media.
