AI 中文总结
本文针对含p-拉普拉斯算子的模型问题，研究反最大值原理的相关性质，包括其一致性、端点映射的连续性等，部分结果针对线性情形，还提出了相关开放问题。
AI 中文摘要
针对有界光滑区域Ω内满足零狄利克雷边界条件的模型问题−Δₚu = λ|u|^(p−2)u + f（其中源函数f非平凡、非负且足够正则），本文给出了关于反最大值原理（AMP）的若干注记。记λ_f为AMP的有效端点，即对任意λ∈(λ₁,λ_f)，问题的每个解在Ω内均为负。本文讨论内容包括：识别使AMP一致的一类源、映射f↦λ_f的下半连续性、λ_f的界、当λ足够大时负解的不存在性（扩展AMP）、反比较原理，以及源的正则性从勒贝格空间到莫雷空间的弱化。部分结果仅针对线性情形p=2，同时作为讨论的一部分，本文还给出了几个相关的开放问题。
英文摘要
We present several observations on the antimaximum principle (AMP) for the model problem $-Δ_p u = λ|u|^{p-2} u + f$ in a bounded smooth domain $Ω$, subject to the zero Dirichlet boundary conditions, and where the source function $f$ is nontrivial, nonnegative, and sufficiently regular. Denote by $λ_f$ the endpoint of validity of the AMP, so that every solution of the problem is negative in $Ω$ for any $λ\in (λ_1,λ_f)$. Our discussion covers the following aspects: identification of a class of sources over which the AMP is uniform, lower semicontinuity of the map $f \mapsto λ_f$, bounds on $λ_f$, the nonexistence of negative solutions for sufficiently large $λ$ (extended AMP), the anticomparison principle, and the weakening of the source regularity from the Lebesgue to Morrey spaces. Some of the results are stated only in the linear case $p=2$. As a part of the discussion, we provide a few related open problems.
Comments19 pages
