  We solve the existence problem for the minimal positive solutions $u\in
L^{p}(\Omega, dx)$ to the Dirichlet problems for sublinear elliptic equations
of the form \[ \begin{cases} Lu=\sigma u^q+\mu\qquad \quad \text{in} \quad
\Omega, \\ \liminf\limits_{x \rightarrow y}u(x) = 0 \qquad y \in
\partial_{\infty}\Omega, \end{cases} \] where $0<q<1$ and $Lu:=-\text{div}
(\mathcal{A}(x)\nabla u)$ is a linear uniformly elliptic operator with bounded
measurable coefficients. The coefficient $\sigma$ and data $\mu$ are
nonnegative Radon measures on an arbitrary domain $\Omega \subset \mathbb{R}^n$
with a positive Green function associated with $L$. Our techniques are based on
the use of sharp Green potential pointwise estimates, weighted norm
inqualities, and norm estimates in terms of generalized energy.
