{"title":"穿刺球中全非线性奇异或退化算子的主特征值","authors":"Françoise Demengel","doi":"10.1016/j.nonrwa.2024.104142","DOIUrl":null,"url":null,"abstract":"<div><p>This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear degenerate or singular uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions <span><math><mrow><mo>(</mo><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>γ</mi></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>)</mo></mrow></math></span> of the equation <span><math><mrow><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup><mi>F</mi><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>)</mo></mrow><mo>+</mo><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>γ</mi></mrow></msub><mfrac><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow></msubsup></mrow><mrow><msup><mrow><mi>r</mi></mrow><mrow><mi>γ</mi></mrow></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><mi>B</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>=</mo><mn>0</mn><mspace></mspace><mtext>on</mtext><mspace></mspace><mi>∂</mi><mi>B</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> in <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>α</mi><mo>></mo><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>γ</mi><mo>></mo><mn>0</mn></mrow></math></span>. We prove existence of radial solutions which are continuous on <span><math><mover><mrow><mi>B</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>¯</mo></mover></math></span> in the case <span><math><mrow><mi>γ</mi><mo><</mo><mn>2</mn><mo>+</mo><mi>α</mi></mrow></math></span>, and a non existence result for <span><math><mrow><mi>γ</mi><mo>></mo><mn>2</mn><mo>+</mo><mi>α</mi></mrow></math></span>. We also give the explicit value of <span><math><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mn>2</mn><mo>+</mo><mi>α</mi></mrow></msub></math></span> in the case of the Pucci’s operators, which generalizes the Hardy–Sobolev constant for the Laplacian, and the previous results of Birindelli et al. <span>[1]</span>.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Principal eigenvalues for Fully Non Linear singular or degenerate operators in punctured balls\",\"authors\":\"Françoise Demengel\",\"doi\":\"10.1016/j.nonrwa.2024.104142\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear degenerate or singular uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions <span><math><mrow><mo>(</mo><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>γ</mi></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>)</mo></mrow></math></span> of the equation <span><math><mrow><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup><mi>F</mi><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>)</mo></mrow><mo>+</mo><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>γ</mi></mrow></msub><mfrac><mrow><msubsup><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow></msubsup></mrow><mrow><msup><mrow><mi>r</mi></mrow><mrow><mi>γ</mi></mrow></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><mi>B</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>∖</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>=</mo><mn>0</mn><mspace></mspace><mtext>on</mtext><mspace></mspace><mi>∂</mi><mi>B</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math></span> in <span><math><mrow><mi>B</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>α</mi><mo>></mo><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>γ</mi><mo>></mo><mn>0</mn></mrow></math></span>. We prove existence of radial solutions which are continuous on <span><math><mover><mrow><mi>B</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>¯</mo></mover></math></span> in the case <span><math><mrow><mi>γ</mi><mo><</mo><mn>2</mn><mo>+</mo><mi>α</mi></mrow></math></span>, and a non existence result for <span><math><mrow><mi>γ</mi><mo>></mo><mn>2</mn><mo>+</mo><mi>α</mi></mrow></math></span>. We also give the explicit value of <span><math><msub><mrow><mover><mrow><mi>λ</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mn>2</mn><mo>+</mo><mi>α</mi></mrow></msub></math></span> in the case of the Pucci’s operators, which generalizes the Hardy–Sobolev constant for the Laplacian, and the previous results of Birindelli et al. <span>[1]</span>.</p></div>\",\"PeriodicalId\":49745,\"journal\":{\"name\":\"Nonlinear Analysis-Real World Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-06-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Real World Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1468121824000828\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Real World Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1468121824000828","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Principal eigenvalues for Fully Non Linear singular or degenerate operators in punctured balls
This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear degenerate or singular uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions of the equation where in , and . We prove existence of radial solutions which are continuous on in the case , and a non existence result for . We also give the explicit value of in the case of the Pucci’s operators, which generalizes the Hardy–Sobolev constant for the Laplacian, and the previous results of Birindelli et al. [1].
期刊介绍:
Nonlinear Analysis: Real World Applications welcomes all research articles of the highest quality with special emphasis on applying techniques of nonlinear analysis to model and to treat nonlinear phenomena with which nature confronts us. Coverage of applications includes any branch of science and technology such as solid and fluid mechanics, material science, mathematical biology and chemistry, control theory, and inverse problems.
The aim of Nonlinear Analysis: Real World Applications is to publish articles which are predominantly devoted to employing methods and techniques from analysis, including partial differential equations, functional analysis, dynamical systems and evolution equations, calculus of variations, and bifurcations theory.