{"title":"Nonlocal Hénon type problem with nonlinearities involving slightly subcritical growth","authors":"Imene Bendahou , Zied Khemiri , Fethi Mahmoudi","doi":"10.1016/j.jde.2024.09.016","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the existence of solutions for the following nonlocal superlinear elliptic problem<span><span><span>(0.1)</span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>β</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi><mo>−</mo><mi>ε</mi></mrow></msup><mspace></mspace></mtd><mtd><mtext>in </mtext><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mtext>in </mtext><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>﹨</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></span></span></span> where <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mi>n</mi><mo>></mo><mn>2</mn><mi>s</mi><mo>,</mo><mi>p</mi><mo>:</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn><mi>s</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>s</mi></mrow></mfrac></math></span> is the Sobolev critical exponent, <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is a smooth bounded domain with Lipschitz boundary, <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span> is the fractional Laplace operator and <span><math><mi>β</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span> is a bounded positive continuous function. We assume that there exists a nondegenerate critical point <span><math><msup><mrow><mi>ξ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>∈</mo><mo>∂</mo><mi>Ω</mi></math></span> of the restriction of <em>β</em> to the boundary ∂Ω such that<span><span><span><math><mrow><mi>∇</mi><mo>(</mo><mi>β</mi><msup><mrow><mo>(</mo><msup><mrow><mi>ξ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>s</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></mfrac></mrow></msup><mo>)</mo><mo>⋅</mo><mi>η</mi><mo>(</mo><msup><mrow><mi>ξ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo><mo>></mo><mn>0</mn><mo>.</mo></mrow></math></span></span></span> Given any integer <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, we show that for <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> small enough, problem <span><span>(0.1)</span></span> has a positive solution, which is a sum of <em>k</em> bubbles which concentrate at <span><math><msup><mrow><mi>ξ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> as <em>ε</em> tends to zero. Also, we prove the existence of nodal (sign changing) solution whose shape resembles a sum of a positive bubble and a negative bubble near the point <span><math><msub><mrow><mi>ξ</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span>. This work can be seen as a nonlocal analogue of the result by Dávila, Faya and Mahmoudi, see <span><span>[28]</span></span>.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039624005953","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the existence of solutions for the following nonlocal superlinear elliptic problem(0.1) where is the Sobolev critical exponent, is a smooth bounded domain with Lipschitz boundary, is the fractional Laplace operator and is a bounded positive continuous function. We assume that there exists a nondegenerate critical point of the restriction of β to the boundary ∂Ω such that Given any integer , we show that for small enough, problem (0.1) has a positive solution, which is a sum of k bubbles which concentrate at as ε tends to zero. Also, we prove the existence of nodal (sign changing) solution whose shape resembles a sum of a positive bubble and a negative bubble near the point . This work can be seen as a nonlocal analogue of the result by Dávila, Faya and Mahmoudi, see [28].
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics