Monotonicity in Half-spaces for p-Laplace Problems with a Sublinear Nonlinearity

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis Pub Date : 2024-07-18 DOI:10.1007/s11118-024-10157-1

Phuong Le

引用次数: 0

Abstract

We prove the monotonicity of positive solutions to the equation \(-\Delta _p u = f(u)\) in \(\mathbb {R}^N_+\) with zero Dirichlet boundary condition, where \(1<p<2\) and \(f:[0,+\infty )\rightarrow \mathbb {R}\) is a continuous function which is positive and locally Lipschitz continuous in \((0,+\infty )\) and \(\liminf _{t\rightarrow 0^+}\frac{f(t)}{t^{p-1}}>0\). Furthermore, we allow f to be sign-changing in the case \(\frac{2N+2}{N+2}<p<2\). The celebrated moving plane method will be used in the proofs of our results.

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具有亚线性非线性的 p-Laplace 问题的半空间单调性

我们证明了方程 \(-\Delta _p u = f(u)\)的正解的单调性，其中 \(1<p<2\) 和 \(f. [0,+\infty )\rightarrow \mathbb {R}^N_+\) 是在（(0,+\infty )）中正且局部 Lipschitz 连续的连续函数：((0,+\infty)\rightarrow\mathbb{R}\)是一个连续函数，在\((0,+\infty)\)和\(\liminf _{t\rightarrow 0^+}\frac{f(t)}{t^{p-1}}>0\)中是正的和局部利普希兹连续的。此外，在 \(\frac{2N+2}{N+2}<p<2\) 的情况下，我们允许 f 是符号变化的。我们将用著名的移动平面法来证明我们的结果。

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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来源期刊

Potential Analysis 数学-数学

CiteScore

2.20

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original papers dealing with potential theory and its applications, probability theory, geometry and functional analysis and in particular estimations of the solutions of elliptic and parabolic equations; analysis of semi-groups, resolvent kernels, harmonic spaces and Dirichlet forms; Markov processes, Markov kernels, stochastic differential equations, diffusion processes and Levy processes; analysis of diffusions, heat kernels and resolvent kernels on fractals; infinite dimensional analysis, Gaussian analysis, analysis of infinite particle systems, of interacting particle systems, of Gibbs measures, of path and loop spaces; connections with global geometry, linear and non-linear analysis on Riemannian manifolds, Lie groups, graphs, and other geometric structures; non-linear or semilinear generalizations of elliptic or parabolic equations and operators; harmonic analysis, ergodic theory, dynamical systems; boundary value problems, Martin boundaries, Poisson boundaries, etc.