Boundary Value Problems最新文献_第3页

Nonexistence of interior bubbling solutions for slightly supercritical elliptic problems 微超临界椭圆型问题内泡解的不存在性

4区数学

Boundary Value Problems Pub Date : 2023-09-11 DOI: 10.1186/s13661-023-01779-2

Mohamed Ben Ayed, Khalil El Mehdi

{"title":"Nonexistence of interior bubbling solutions for slightly supercritical elliptic problems","authors":"Mohamed Ben Ayed, Khalil El Mehdi","doi":"10.1186/s13661-023-01779-2","DOIUrl":"https://doi.org/10.1186/s13661-023-01779-2","url":null,"abstract":"Abstract In this paper, we consider the Neumann elliptic problem $(mathcal{P}_{varepsilon})$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:math> : $-Delta u +mu u = u^{(({n+2})/({n-2}))+varepsilon}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>μ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msup> </mml:math> , $u>0$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>u</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> in Ω, ${partial u}/{partial nu}=0$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>∂</mml:mi> <mml:mi>u</mml:mi> <mml:mo>/</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> on ∂ Ω, where Ω is a smooth bounded domain in $mathbb{R}^{n}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , $ngeq 4$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:math> , ε is a small positive real, and μ is a fixed positive number. We show that, in contrast with the three dimensional case, $(mathcal{P}_{varepsilon})$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:math> has no solution blowing up at only interior points as ε goes to zero. The proof strategy consists in testing the equation by appropriate vector fields and then using refined asymptotic estimates in the neighborhood of bubbles, we obtain equilibrium conditions satisfied by the concentration parameters. The careful analysis of these balancing conditions allows us to obtain our results.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135980346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}