{"title":"混合局部和非局部超临界狄利克雷问题","authors":"David Amundsen, Abbas Moameni, Remi Yvant Temgoua","doi":"10.3934/cpaa.2023104","DOIUrl":null,"url":null,"abstract":"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<q<2<p $ has a positive solution as well as a sequence of sign-changing solutions with a negative energy for small values of $ \\mu $. Here $ u $ can be either a scalar function, or a vector valued function so that (1) turns into a system with supercritical nonlinearity. Moreover, whenever the domain is symmetric, we also prove the existence of symmetric solutions enjoying the same symmetry properties. We shall also prove an existence result for the supercritical Hamiltonian system$ Lu = |v|^{p-2}v, \\qquad Lv = |u|^{d-2}u+\\mu |u|^{q-2}u $with the Dirichlet boundary condition on $ \\Omega $ where $ 1<q<2<p, d $. Our method is variational, and in both problems the lack of compactness for the supercritical problem is recovered by working on a closed convex subset of an appropriate function space.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"47 1","pages":"0"},"PeriodicalIF":1.0000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Mixed local and nonlocal supercritical Dirichlet problems\",\"authors\":\"David Amundsen, Abbas Moameni, Remi Yvant Temgoua\",\"doi\":\"10.3934/cpaa.2023104\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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<q<2<p $ has a positive solution as well as a sequence of sign-changing solutions with a negative energy for small values of $ \\\\mu $. Here $ u $ can be either a scalar function, or a vector valued function so that (1) turns into a system with supercritical nonlinearity. Moreover, whenever the domain is symmetric, we also prove the existence of symmetric solutions enjoying the same symmetry properties. We shall also prove an existence result for the supercritical Hamiltonian system$ Lu = |v|^{p-2}v, \\\\qquad Lv = |u|^{d-2}u+\\\\mu |u|^{q-2}u $with the Dirichlet boundary condition on $ \\\\Omega $ where $ 1<q<2<p, d $. Our method is variational, and in both problems the lack of compactness for the supercritical problem is recovered by working on a closed convex subset of an appropriate function space.\",\"PeriodicalId\":10643,\"journal\":{\"name\":\"Communications on Pure and Applied Analysis\",\"volume\":\"47 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Pure and Applied Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/cpaa.2023104\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/cpaa.2023104","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
摘要
本文研究了一类具有超临界非线性的混合局部和非局部狄利克雷问题。我们首先建立了问题$ \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。
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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