分数阶拉普拉斯非线性方程的尖锐临界指数

IF 0.6 4区 数学 Q3 MATHEMATICS
Zixia Yuan, Zimin Tang
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引用次数: 0

摘要

本文考虑两类具有分数阶拉普拉斯算子的非线性偏微分方程,即(−Δ)α2(um)=u|u|q−1+w(x),x∈RN,1≤mα和∂ku∂tk+(−Δ)α2u=uq,(x,t)∈rnx(0,+∞),k≥1,0α。对于第一个方程的所有x∈RN定义的解称为完整解,对于第二个方程的所有(x,t)∈rnx[0,+∞)定义的解称为全局解。在不同的q范围上建立了若干存在性和不存在性定理,从而得到了这些方程解的存在性和不存在性与非线性项中指标q的各自关系。结果表明,在m = 1和k = 1的情况下,我们的结果是清晰的。此外,我们证明了我们构造的m = 1与非齐次项w相关的第一个方程解的正、对称和正则性。关键词:分数阶拉普拉斯临界指数存在性存在性ams主题分类:35B0835B3335R11披露声明作者未报告潜在的利益冲突。本研究得到四川省中央政府地方科技发展专项项目[批准号2021ZYD0014]的部分支持。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Sharp critical exponents for nonlinear equations with the fractional Laplacian
In this paper we consider two classes of nonlinear partial differential equations with the fractional Laplacian, namely (−Δ)α2(um)=u|u|q−1+w(x),x∈RN,1≤mα and ∂ku∂tk+(−Δ)α2u=uq,(x,t)∈RN×(0,+∞),k≥1, 0<α≤2, N>α. Solutions defined for all x∈RN of the first equation are referred to as entire solutions, while solutions defined for all (x,t)∈RN×[0,+∞) of the second equation are referred to as global solutions. Several existence and nonexistence theorems are established over different ranges of q, and thus the respective relations between the existence, nonexistence of solutions for these equations and the index q in the nonlinear terms are obtained. It is illustrated that our results are sharp in cases of m = 1 and k = 1 respectively. In addition, we prove the positivity, symmetry and odevity of solutions we constructed for the first equation with m = 1 associated with the inhomogeneous term w.KEYWORDS: Fractional Laplaciancritical exponentexistencenonexistenceAMS SUBJECT CLASSIFICATIONS: 35B0835B3335R11 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was partially supported by the Special Project for Local Science and Technology Development Guided by the Central Government of Sichuan Province [grant number 2021ZYD0014].
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来源期刊
CiteScore
2.00
自引率
11.10%
发文量
97
审稿时长
6-12 weeks
期刊介绍: Complex Variables and Elliptic Equations is devoted to complex variables and elliptic equations including linear and nonlinear equations and systems, function theoretical methods and applications, functional analytic, topological and variational methods, spectral theory, sub-elliptic and hypoelliptic equations, multivariable complex analysis and analysis on Lie groups, homogeneous spaces and CR-manifolds. The Journal was formally published as Complex Variables Theory and Application.
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