Harnack inequality for non-divergence structure semi-linear elliptic equations
In this paper, the authors prove a Harnack inequality for non-negative solutions of \(Lu = f(u)\), where \(L\) is a non-divergence structure uniformly elliptic operator and \(f\) is a non-decreasing function that satisfies an appropriate growth conditions at infinity. In [Trans. Am. Math. Soc. 363, No. 10, 5131--5178 (2011; Zbl 1225.35076)], \textit{M. Dindoš} proved a Harnack inequality for the solutions of the equation \(\Delta u = f(u)\), where \(f: \mathbb{R}^+ \rightarrow \mathbb{R}^+ \) is non-decreasing and continuous such that \(f(t) > 0\) for \(t > 0\), and there is \(\theta > 1\) such that \(\underset{t\rightarrow \infty }\liminf \frac{f(\theta t)}{\theta f(t)} >1.\) In this paper, the authors extend Dindoš' Harnack inequality results to more general operators and provide an example to show that Dindoš' condition cannot be too much relaxed for the stated Harnack inequality to hold.
non-divergence structure elliptic operator
Alexandroff-Bakelman-Pucci maximum principle
scientific article; zbMATH DE number 6919837
scientific article