树上半线性椭圆方程正解的存在性和渐近行为

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations Pub Date : 2024-10-17 DOI:10.1016/j.jde.2024.10.009

Yating Niu , Yingshu Lü

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引用次数: 0

摘要

设 G=(V,E) 是局部有限树，Δ 是归一化拉普拉卡。在本文中，我们考虑 G 上的以下半线性方程 (0.1)Δu+f(u)=0。我们首先在 f 的一般假设下建立了 (0.1) 正解的存在性和不存在性，然后找到了规则树上 (0.1) 的临界指数。此外，我们还证明了径向解的一些有趣性质，以及在更一般的 f 条件下径向解的渐近行为。最后，不存在结果可以推广到加权树上的椭圆系统。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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

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)

Δ u + f (u) = 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.

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来源期刊

Journal of Differential Equations 数学-数学

CiteScore

4.40

自引率

8.30%

发文量

543

审稿时长

9 months

期刊介绍： The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics