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

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 $$\frac{1}{c}H_D^\varphi f\leq H_D^\psi f\leq c H_D^\varphi f.$$