{"title":"Asymptotic decay of solutions for sublinear fractional Choquard equations","authors":"Marco Gallo","doi":"10.1016/j.na.2024.113515","DOIUrl":null,"url":null,"abstract":"<div><p>Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation <span><span><span><math><mrow><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>+</mo><mi>μ</mi><mi>u</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>∗</mo><mi>F</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mtext>on</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span></span></span>where <span><math><mrow><mi>s</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>N</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>μ</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> denotes the Riesz potential and <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mi>f</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mi>d</mi><mi>τ</mi></mrow></math></span> is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than <span><math><mrow><mo>∼</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mi>N</mi><mo>+</mo><mn>2</mn><mi>s</mi></mrow></msup></mrow></mfrac></mrow></math></span>. The result is new even for homogeneous functions <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow></math></span>, <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>[</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mi>α</mi></mrow><mrow><mi>N</mi></mrow></mfrac><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>, and it complements the decays obtained in the linear and superlinear cases in Cingolani et al. (2022); D’Avenia et al. (2015). Differently from the local case <span><math><mrow><mi>s</mi><mo>=</mo><mn>1</mn></mrow></math></span> in Moroz and Van Schaftingen (2013), new phenomena arise connected to a new “<span><math><mi>s</mi></math></span>-sublinear” threshold that we detect on the growth of <span><math><mi>f</mi></math></span>. To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"242 ","pages":"Article 113515"},"PeriodicalIF":1.3000,"publicationDate":"2024-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24000348/pdfft?md5=d14fa534745d380d224b53616b67a72e&pid=1-s2.0-S0362546X24000348-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24000348","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
Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation where , , , , denotes the Riesz potential and is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than . The result is new even for homogeneous functions , , and it complements the decays obtained in the linear and superlinear cases in Cingolani et al. (2022); D’Avenia et al. (2015). Differently from the local case in Moroz and Van Schaftingen (2013), new phenomena arise connected to a new “-sublinear” threshold that we detect on the growth of . To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers.
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