Minimal Lp-solutions to singular sublinear elliptic problems

Abstract

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 $\left(\begin{array}{c}Lu=\sigma u^q+\mu \text{in}\Omega ,\\ \underset{x\to y}{lim\; inf}u\left(x\right)=0y\in {\partial }_{\infty }\Omega ,\end{array}\right)$where $0<q<1$ and $Lu≔-\text{div}(A\left(x\right)\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients. The coefficient $\sigma$ and data $\mu$ are nonnegative Radon measures on an arbitrary domain $\Omega \subset R^n$ with a positive Green function associated with $L$. Our techniques are based on the use of sharp Green potential pointwise estimates, weighted norm inequalities, and norm estimates in terms of generalized energy.