Comparison of harmonic kernels associated with a class of semilinear elliptic equations (Q2879478)

In this paper the following semilinear elliptic equation is studied NEWLINE\[NEWLINE \begin{cases}\Delta u=2\varphi(u) & \text{ in }D, \\ u=f & \text{ on } \partial D.\end{cases} NEWLINE\]NEWLINE Here \(D\) is a bounded smooth domain in \(\mathbb{R}^N\) with \(N\geq 3\) and \(f\) is a positive continuous function on \(\partial D\).NEWLINENEWLINEUnder the condition that \(\varphi: \mathbb{R}_+\rightarrow \mathbb{R}_+\) is a continuous nondecreasing function with \(\varphi(0)=0\), the equation admits a unique solution \(u\in {\mathcal C}^+(\overline{D})\), which is denoted \(H_D^{\varphi}f\). The main purpose of the paper is to establish the proportionality of two such solutions. Given two functions \(\varphi\) and \(\psi\), \(H_D^{\varphi}f\) and \(H_D^{\psi}f\) are said to be proportional and is written \(H_D^{\varphi}f \approx H_D^{\psi}f\) if there exists \(c>0\) such that NEWLINE\[NEWLINE{1\over c}H_D^{\psi}f\leq H_D^{\varphi}f\leq cH_D^{\psi}fNEWLINE\]NEWLINE on \(D\).NEWLINENEWLINEThe main result of the paper is the following: If \(\displaystyle \limsup_{t\rightarrow 0}{\varphi(t)\over t}<\infty\), then \(H_D^{\varphi}f\approx H_D^{0}f\) for every \(f\in {\mathcal C}^+(\partial D)\). On the other hand, if there exists \(\epsilon >0\) such that NEWLINE\[NEWLINE \int_0^{\epsilon}\left(\int_0^s\varphi(r)dr\right)^{-1/2}ds<\infty,NEWLINE\]NEWLINE then there exists \(f\in {\mathcal C}^+(\partial D)\) such that \(H_D^{\varphi}f \not\approx H_D^{0}f\). A special case of the second part of the result gives an affirmatively answer to a conjecture contained in [\textit{R. Atar} et al., Electron. J. Probab. 14, 50--71 (2009; Zbl 1190.60056)]. The conjecture states that for \(0<p<1\), there exists \(f\in {\mathcal C}^+(\partial D)\) such that \(H_D^{t^p}f \not\approx H_D^{0}f\).