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{"title":"具有 k-Hessian 算子和 Lane-Emden 型非线性的多参数 Dirichlet 系统的 k-convex 解法","authors":"Xingyue He, Chenghua Gao, Jingjing Wang","doi":"10.1515/anona-2023-0136","DOIUrl":null,"url":null,"abstract":"\n <jats:p>In this article, our main aim is to investigate the existence of radial <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0136_eq_001.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>k</m:mi>\n </m:math>\n <jats:tex-math>k</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>-convex solutions for the following Dirichlet system with <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0136_eq_002.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>k</m:mi>\n </m:math>\n <jats:tex-math>k</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>-Hessian operators: <jats:disp-formula id=\"j_anona-2023-0136_eq_001\">\n <jats:alternatives>\n <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0136_eq_003.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:msub>\n <m:mrow>\n <m:mi>S</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>k</m:mi>\n </m:mrow>\n </m:msub>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:msup>\n <m:mrow>\n <m:mi>D</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:msup>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mo>=</m:mo>\n <m:msub>\n <m:mrow>\n <m:mi>λ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mn>1</m:mn>\n </m:mrow>\n </m:msub>\n <m:msub>\n <m:mrow>\n <m:mi>ν</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mn>1</m:mn>\n </m:mrow>\n </m:msub>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mo>∣</m:mo>\n <m:mi>x</m:mi>\n <m:mo>∣</m:mo>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:msup>\n <m:mrow>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mo>−</m:mo>\n <m:mi>u</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n </m:mrow>\n <m:mrow>\n <m:msub>\n <m:mrow>\n <m:mi>p</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mn>1</m:mn>\n </m:mrow>\n </m:msub>\n </m:mrow>\n </m:msup>\n <m:msup>\n <m:mrow>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mo>−</m:mo>\n <m:mi>v</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n </m:mrow>\n <m:mrow>\n <m:msub>\n <m:mrow>\n <m:mi>q</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mn>1</m:mn>\n </m:mrow>\n </m:msub>\n </m:mrow>\n </m:msup>\n </m:mtd>\n <m:mtd columnalign=\"left\">\n <m:mi mathvariant=\"normal\">in</m:mi>\n <m:mspace width=\"1em\" />\n <m:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">ℬ</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi>R</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mo>,</m:mo>\n </m:mtd>\n </m:mtr>\n <m:mtr>\n <m:mtd columnalign=\"left\">\n <m:msub>\n <m:mrow>\n <m:mi>S</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>k</m:mi>\n </m:mrow>\n </m:msub>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:msup>\n <m:mrow>\n <m:mi>D</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:msup>\n <m:mi>v</m:mi>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n <m:mo>=</m:mo>\n <m:msub>\n <m:mrow>\n <m:mi>λ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mn>2</m:mn>\n ","PeriodicalId":3,"journal":{"name":"ACS Applied Electronic Materials","volume":null,"pages":null},"PeriodicalIF":4.3000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities\",\"authors\":\"Xingyue He, Chenghua Gao, Jingjing Wang\",\"doi\":\"10.1515/anona-2023-0136\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n <jats:p>In this article, our main aim is to investigate the existence of radial <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0136_eq_001.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>k</m:mi>\\n </m:math>\\n <jats:tex-math>k</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula>-convex solutions for the following Dirichlet system with <jats:inline-formula>\\n <jats:alternatives>\\n <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0136_eq_002.png\\\" />\\n <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\">\\n <m:mi>k</m:mi>\\n </m:math>\\n <jats:tex-math>k</jats:tex-math>\\n </jats:alternatives>\\n </jats:inline-formula>-Hessian operators: <jats:disp-formula id=\\\"j_anona-2023-0136_eq_001\\\">\\n <jats:alternatives>\\n <jats:graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_anona-2023-0136_eq_003.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:msub>\\n <m:mrow>\\n <m:mi>S</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mi>k</m:mi>\\n </m:mrow>\\n </m:msub>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:msup>\\n <m:mrow>\\n <m:mi>D</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mn>2</m:mn>\\n </m:mrow>\\n </m:msup>\\n <m:mi>u</m:mi>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n <m:mo>=</m:mo>\\n <m:msub>\\n <m:mrow>\\n <m:mi>λ</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mn>1</m:mn>\\n </m:mrow>\\n </m:msub>\\n <m:msub>\\n <m:mrow>\\n <m:mi>ν</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mn>1</m:mn>\\n </m:mrow>\\n </m:msub>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:mo>∣</m:mo>\\n <m:mi>x</m:mi>\\n <m:mo>∣</m:mo>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n <m:msup>\\n <m:mrow>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:mo>−</m:mo>\\n <m:mi>u</m:mi>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n </m:mrow>\\n <m:mrow>\\n <m:msub>\\n <m:mrow>\\n <m:mi>p</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mn>1</m:mn>\\n </m:mrow>\\n </m:msub>\\n </m:mrow>\\n </m:msup>\\n <m:msup>\\n <m:mrow>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:mo>−</m:mo>\\n <m:mi>v</m:mi>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n </m:mrow>\\n <m:mrow>\\n <m:msub>\\n <m:mrow>\\n <m:mi>q</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mn>1</m:mn>\\n </m:mrow>\\n </m:msub>\\n </m:mrow>\\n </m:msup>\\n </m:mtd>\\n <m:mtd columnalign=\\\"left\\\">\\n <m:mi mathvariant=\\\"normal\\\">in</m:mi>\\n <m:mspace width=\\\"1em\\\" />\\n <m:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">ℬ</m:mi>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:mi>R</m:mi>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n <m:mo>,</m:mo>\\n </m:mtd>\\n </m:mtr>\\n <m:mtr>\\n <m:mtd columnalign=\\\"left\\\">\\n <m:msub>\\n <m:mrow>\\n <m:mi>S</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mi>k</m:mi>\\n </m:mrow>\\n </m:msub>\\n <m:mrow>\\n <m:mo>(</m:mo>\\n <m:mrow>\\n <m:msup>\\n <m:mrow>\\n <m:mi>D</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mn>2</m:mn>\\n </m:mrow>\\n </m:msup>\\n <m:mi>v</m:mi>\\n </m:mrow>\\n <m:mo>)</m:mo>\\n </m:mrow>\\n <m:mo>=</m:mo>\\n <m:msub>\\n <m:mrow>\\n <m:mi>λ</m:mi>\\n </m:mrow>\\n <m:mrow>\\n <m:mn>2</m:mn>\\n \",\"PeriodicalId\":3,\"journal\":{\"name\":\"ACS Applied Electronic Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.3000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Electronic 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