B. Ricceri
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{"title":"Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions","authors":"B. Ricceri","doi":"10.1515/anona-2023-0104","DOIUrl":null,"url":null,"abstract":"\n <jats:p>Let <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0104_eq_001.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi mathvariant=\"normal\">Ω</m:mi>\n <m:mo>⊂</m:mo>\n <m:msup>\n <m:mrow>\n <m:mi mathvariant=\"bold\">R</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>n</m:mi>\n </m:mrow>\n </m:msup>\n </m:math>\n <jats:tex-math>\\Omega \\subset {{\\bf{R}}}^{n}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0104_eq_002.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>q</m:mi>\n <m:mo>∈</m:mo>\n <m:mrow>\n <m:mo stretchy=\"false\">]</m:mo>\n <m:mrow>\n <m:mn>0</m:mn>\n <m:mo>,</m:mo>\n <m:mn>1</m:mn>\n </m:mrow>\n <m:mo stretchy=\"false\">[</m:mo>\n </m:mrow>\n </m:math>\n <jats:tex-math>q\\in ]0,1{[}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>, <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0104_eq_003.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>α</m:mi>\n <m:mo>∈</m:mo>\n <m:msup>\n <m:mrow>\n <m:mi>L</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>∞</m:mi>\n </m:mrow>\n </m:msup>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi mathvariant=\"normal\">Ω</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n </m:math>\n <jats:tex-math>\\alpha \\in {L}^{\\infty }\\left(\\Omega )</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>, with <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0104_eq_004.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>α</m:mi>\n <m:mo>></m:mo>\n <m:mn>0</m:mn>\n </m:math>\n <jats:tex-math>\\alpha \\gt 0</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>, and <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0104_eq_005.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>k</m:mi>\n <m:mo>∈</m:mo>\n <m:mi mathvariant=\"bold\">N</m:mi>\n </m:math>\n <jats:tex-math>k\\in {\\bf{N}}</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>. Then, the problem <jats:disp-formula id=\"j_anona-2023-0104_eq_001\">\n <jats:alternatives>\n <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0104_eq_006.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\n <m:mfenced open=\"{\" close=\"\">\n <m:mrow>\n <m:mtable displaystyle=\"true\">\n <m:mtr>\n <m:mtd columnalign=\"left\">\n <m:mo>−</m:mo>\n <m:mi>tan</m:mi>\n <m:mfenced open=\"(\" close=\")\">\n <m:mrow>\n <m:munder>\n <m:mrow>\n <m:mstyle displaystyle=\"true\">\n <m:mo>∫</m:mo>\n </m:mstyle>\n </m:mrow>\n <m:mrow>\n <m:mi mathvariant=\"normal\">Ω</m:mi>\n </m:mrow>\n </m:munder>\n <m:mo>∣</m:mo>\n <m:mrow>\n <m:mo>∇</m:mo>\n </m:mrow>\n <m:mi>u</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>x</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:msup>\n <m:mrow>\n <m:mo>∣</m:mo>\n </m:mrow>\n <m:mrow>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:msup>\n <m:mi mathvariant=\"normal\">d</m:mi>\n <m:mi>x</m:mi>\n </m:mrow>\n </m:mfenced>\n <m:mi mathvariant=\"normal\">Δ</m:mi>\n <m:mi>u</m:mi>\n <m:mo>=</m:mo>\n <m:mi>α</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>x</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:msup>\n <m:mrow>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>q</m:mi>\n </m:mrow>\n </m:msup>\n <m:mspace width=\"1.0em\" />\n </m:mtd>\n <m:mtd columnalign=\"left\">\n <m:mstyle>\n <m:mspace width=\"0.1em\" />\n <m:mtext>in</m:mtext>\n <m:mspace width=\"0.1em\" />\n </m:mstyle>\n <m:mspace width=\"0.33em\" />\n <m:mi mathvariant=\"normal\">Ω</m:mi>\n </m:mtd>\n </m:mtr>\n <m:mtr>\n <m:mtd columnalign=\"left\">\n <m:mi>u</m:mi>\n <m:mo>></m:mo>\n <m:mn>0</m:mn>\n <m:mspace width=\"1.0em\" />\n </m:mtd>\n <m:mtd columnalign=\"left\">\n <m:mstyle>\n <m:mspace width=\"0.1em\" />\n <m:mtext>in</m:mtext>\n ","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/anona-2023-0104","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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让 Ω ⊂ R n\Omega \子集 {{\bf{R}}}^{n} 是一个光滑有界域。本文将证明一个结果,下面是其副产品:设 q∈ ] 0 , 1 [ q\in ]0,1{[} , α ∈ L ∞ ( Ω ) \α > 0 \alpha \gt 0 , and k ∈ N k\in {bf{N}}. .
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