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

Abstract

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

15 pages