Estimates of Nonnegative Solutions to Semilinear Elliptic Equations

TL;DR

This paper establishes bounds for nonnegative solutions to certain semilinear elliptic inequalities using a function determined solely by the nonlinearity, independent of the operator or domain.

Contribution

It introduces a universal function that provides lower bounds for solutions of semilinear elliptic inequalities, independent of the specific elliptic operator or domain.

Findings

Derived a universal function  that bounds solutions from below.

Established bounds applicable to a wide class of elliptic inequalities.

The bounds depend only on the nonlinearity  and the Green function of the source term.

Abstract

Let $L$ be a second order uniformly elliptic differential operator in a domain $D$ of $\mathbb{R}^{d}$, $\psi:\mathbb{R}_+\to \mathbb{R}_+$ be a nondecreasing continuous function and let $\xi,g:D\to\mathbb{R}_+$ be locally bounded Borel measurable functions. Under appropriate conditions, we determine a function $\varphi$ with values in $]0,1]$ such that for every nonnegative solution to inequality $-Lu+\xi\psi(u) \geq g$ in $D$ and for every $x\in D$, $$ u(x)\geq p(x)\,\varphi\left(\frac{G_D(\xi\psi(p))(x)}{p(x)}\right) $$ where $p=G_Dg$ is the Green function of $g$. The function $\varphi$ is completely determined by $\psi$ and does not depend on $L,D,\xi$ or $g$.