混合局部和非局部超临界狄利克雷问题

IF 1 3区数学 Q1 MATHEMATICS

Communications on Pure and Applied Analysis Pub Date : 2023-01-01 DOI:10.3934/cpaa.2023104

David Amundsen, Abbas Moameni, Remi Yvant Temgoua

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引用次数: 2

摘要

本文研究了一类具有超临界非线性的混合局部和非局部狄利克雷问题。我们首先建立了问题$ \begin{equation} Lu = |u|^{p-2}u+\mu |u|^{q-2}u\quad\text{in}\; \; \Omega, \quad\quad u = 0\quad\text{in}\; \; \mathbb{R}^N\setminus\Omega, ~~~(1) \end{equation} $的多重性结果，其中$ s\in(0, 1) $和$ \Omega\subset\mathbb{R}^N $的$ L: = -\Delta +(-\Delta)^s $是一个有界域。准确地说，我们证明了$ 1<q<2<p $的问题(1)有一个正解，以及对于$ \mu $的小值具有负能量的换号解序列。这里$ u $可以是标量函数，也可以是矢量值函数，这样(1)就变成了一个具有超临界非线性的系统。此外，只要定义域是对称的，我们也证明了具有相同对称性的对称解的存在性。我们还证明了在$ \Omega $上具有Dirichlet边界条件的超临界哈密顿系统$ Lu = |v|^{p-2}v, \qquad Lv = |u|^{d-2}u+\mu |u|^{q-2}u $的存在性结果，其中$ 1<q<2<p, d $。我们的方法是变分的，并且在这两个问题中，超临界问题的紧性缺失通过在适当函数空间的闭凸子集上工作来恢复。

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Mixed local and nonlocal supercritical Dirichlet problems

In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem $ \begin{equation} Lu = |u|^{p-2}u+\mu |u|^{q-2}u\quad\text{in}\; \; \Omega, \quad\quad u = 0\quad\text{in}\; \; \mathbb{R}^N\setminus\Omega, ~~~(1) \end{equation} $ where $ L: = -\Delta +(-\Delta)^s $ for $ s\in(0, 1) $ and $ \Omega\subset\mathbb{R}^N $ is a bounded domain. Precisely, we show that problem (1) for $ 1

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来源期刊

Communications on Pure and Applied Analysis 数学-数学

CiteScore

1.90

自引率

10.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： CPAA publishes original research papers of the highest quality in all the major areas of analysis and its applications, with a central theme on theoretical and numeric differential equations. Invited expository articles are also published from time to time. It is edited by a group of energetic leaders to guarantee the journal''s highest standard and closest link to the scientific communities.

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