Quantitative Local Bounds for Subcritical Semilinear Elliptic Equations

TL;DR

This paper establishes explicit local upper and lower bounds for solutions of subcritical semilinear elliptic equations, providing concrete constants and resulting in local Harnack inequalities and gradient bounds.

Contribution

It introduces explicit constants for local bounds of solutions to subcritical semilinear elliptic equations, enhancing understanding of solution behavior without boundary reference.

Findings

Explicit local bounds for solutions derived

Local Harnack inequalities with explicit constants established

Gradient bounds for solutions obtained

Abstract

The purpose of this paper is to prove local upper and lower bounds for weak solutions of semilinear elliptic equations of the form $-\Delta u= c u^p$, with $0<p<p_s=(d+2)/(d-2)$, defined on bounded domains of $\RR^d$, $d\ge 3$, without reference to the boundary behaviour. We give an explicit expression for all the involved constants. As a consequence, we obtain local Harnack inequalities with explicit constant, as well as gradient bounds.