Positive solutions to sublinear elliptic problem

TL;DR

This paper investigates the existence and properties of positive solutions to a class of sublinear elliptic equations involving a second order elliptic operator, providing characterizations based on thinness concepts.

Contribution

It introduces a characterization of the nonlinear term  for which the elliptic problem admits nonnegative bounded solutions, using the notion of thinness.

Findings

Characterization of  ensuring existence of solutions

Use of thinness to analyze solution properties

Conditions for bounded positive solutions

Abstract

Let $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $\Omega $ in $\mathbb{R}^d $, $d\geq3$, such that $L1\leq 0$. We study existence and properties of continuous solutions to the following problem \begin{equation}\label{00} Lu=\varphi(\cdot,u),% & \hbox{in $\Omega $; in the sens of distribution;} \\ \end{equation} in $\Omega,$ where $\Omega$ is a Greenian domain for $L$ {(possibly unbounded)} in $\mathbb{R}^d$ and $\varphi $ is a nonnegative function on $\Omega\times [0,+\infty [$ increasing with respect to the second variable. By means of thinness, we obtain a characterization of $\varphi$ for which \eqref{00} has a nonnegative nontrivial bounded solution.