{"title":"亚线性分数 Choquard 方程解的渐近衰减","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":null,"pages":null},"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":"{\"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\":null,\"pages\":null},\"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}","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
摘要
本文的目标是研究以下双非局部方程 (-Δ)su+μu=(Iα∗F(u))f(u)onRNwhere s∈(0,1), N≥2, α∈(0,N), μ>;0,Iα 表示里兹电势,F(t)=∫0tf(τ)dτ 是一个在原点有亚线性增长的一般非线性。所发现的衰减是多项式类型的,速率可能慢于 ∼1|x|N+2s。即使对于同质函数f(u)=|u|r-2u,r∈[N+αN,2),这一结果也是新的,它补充了Cingolani等人(2022年)和D'Avenia等人(2015年)在线性和超线性情况下获得的衰减。与 Moroz 和 Van Schaftingen (2013)中 s=1 的局部情况不同,新现象的出现与我们检测到的 f 增长的新 "s-次线性 "阈值有关。
Asymptotic decay of solutions for sublinear fractional Choquard equations
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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