Minimal L^p-Solutions to Singular Sublinear Elliptic Problems

We solve the existence problem for the minimal positive solutions $u\in L^p(\mathrm{\Omega },dx)$ to the Dirichlet problems for sublinear elliptic equations of the form $$\{\begin{array}{l}Lu=\sigma u^q+\mu \text{in}\mathrm{\Omega },\\ \underset{x\to y}{\mathrm{lim\hspace{0.17em}inf}}u(x)=0y\in {\mathrm{\partial }}_{\mathrm{\infty }}\mathrm{\Omega },\end{array}$$ where $0<q<1$ and $Lu:=-\text{div}(\mathcal{A}(x)\mathrm{\nabla }u)$ is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient $\sigma$ and data $\mu$ are nonnegative Radon measures on an arbitrary domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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.