The Dirichlet problem for sublinear elliptic equations with source

TL;DR

This paper establishes conditions for the existence and uniqueness of positive solutions to a sublinear elliptic equation with measure data, providing estimates and criteria in a general domain.

Contribution

It introduces necessary and sufficient conditions for solutions with measure data and boundary conditions, including estimates and uniqueness results.

Findings

Characterization of measures for solution existence

Two-sided pointwise estimates for solutions

Uniqueness of positive continuous solutions

Abstract

We present a necessary and sufficient condition on nonnegative Radon measures $\mu$ and $\nu$ for the existence of a positive continuous solution of the Dirichlet problem for the sublinear elliptic equation $-\Delta u=\mu u^q+\nu$ with prescribed nonnegative continuous boundary data in a general domain. Moreover, two-sided pointwise estimates of Brezis-Kamin type for positive bounded solutions and the uniqueness of a positive continuous $L^q$-solution are investigated.