Comparison of harmonic kernels associated to a class of semilinear elliptic equations

TL;DR

This paper investigates the comparison of harmonic kernels associated with semilinear elliptic equations, establishing conditions under which solutions to different nonlinear problems are comparable via a constant factor.

Contribution

It introduces a framework for comparing solutions of semilinear elliptic equations through harmonic kernels and identifies conditions for their bounded ratio.

Findings

Existence of a constant c such that (1/c)H_D^ϕ f ≤ H_D^ψ f ≤ c H_D^ϕ f

Unique solutions to the semilinear elliptic boundary value problem under certain assumptions

Comparison results for harmonic kernels associated with different nonlinearities

Abstract

Let $D$ be a smooth domain in $\mathbb{R}^N$, $N\geq 3$ and let $f$ be a positive continuous function on $\partial D$. Under some assumptions on $\varphi$, it is shown that the problem $\Delta u=2\varphi(u)$ in $D$ and $u=f$ on $\partial D$, admits a unique solution which will be denoted by $H_D^\varphi f$. Given two functions $\varphi$ and $\psi$, our main goal in this paper is to investigate the existence of a constant $c>0$ such that $$\frac{1}{c}H_D^\varphi f\leq H_D^\psi f\leq c H_D^\varphi f.$$