Existence and asymptotic behaviors of positive solutions for a semilinear elliptic equation on trees

Abstract

Let $G=(V,E)$ be a locally finite tree, Δ be the normalized Laplacian. In this paper, we consider the following semilinear equation on G(0.1)$\Delta u+f\left(u\right)=0.$ We first establish the existence and nonexistence of positive solutions to (0.1) with a general assumption on f, and then find the critical exponent for (0.1) on a regular tree. Moreover, we prove some interesting properties of radial solutions and the asymptotic behaviors of radial solutions under a more general condition on f. Finally, the nonexistence results can be generalized to the elliptic system on a weighted tree.