ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS TO FRACTIONAL NEUMANN PROBLEMS

IF 0.5 4区 数学 Q3 MATHEMATICS
SOMNATH GANDAL, JAGMOHAN TYAGI
{"title":"ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS TO FRACTIONAL NEUMANN PROBLEMS","authors":"SOMNATH GANDAL, JAGMOHAN TYAGI","doi":"10.1017/s1446788724000107","DOIUrl":null,"url":null,"abstract":"<p>We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems: <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ \\begin{align*} \\begin{cases} { d(-\\Delta)^{s}u+ u= \\vert u\\vert^{p-1}u } &amp; \\text{in } \\Omega, \\\\ {u&gt;0} &amp; \\text{in } \\Omega, \\\\ { \\mathcal{N}_{s}u=0 } &amp; \\text{in } \\mathbb{R}^{n}\\setminus \\overline{\\Omega}, \\end{cases} \\end{align*} $$</span></span></img></span></p><p>where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\Omega \\subset \\mathbb {R}^{n}$</span></span></img></span></span> is a bounded domain of class <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$C^{1,1}$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$1&lt;p&lt;({n+s})/({n-s}),\\,n&gt;\\max \\{1, 2s \\}, 0&lt;s&lt;1, d&gt;0$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathcal {N}_{s}u$</span></span></img></span></span> is the nonlocal Neumann derivative. We show that for small <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$d,$</span></span></img></span></span> the least energy solutions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$u_d$</span></span></img></span></span> of the above problem achieve an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$L^{\\infty }$</span></span></img></span></span>-bound independent of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$d.$</span></span></img></span></span> Using this together with suitable <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline9.png\"/><span data-mathjax-type=\"texmath\"><span>$L^{r}$</span></span></span></span>-estimates on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline10.png\"/><span data-mathjax-type=\"texmath\"><span>$u_d,$</span></span></span></span> we show that the least energy solution <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline11.png\"/><span data-mathjax-type=\"texmath\"><span>$u_d$</span></span></span></span> achieves a maximum on the boundary of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240913062454564-0570:S1446788724000107:S1446788724000107_inline12.png\"/><span data-mathjax-type=\"texmath\"><span>$\\Omega $</span></span></span></span> for <span>d</span> sufficiently small.</p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"2 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Australian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s1446788724000107","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems: Abstract Image$$ \begin{align*} \begin{cases} { d(-\Delta)^{s}u+ u= \vert u\vert^{p-1}u } & \text{in } \Omega, \\ {u>0} & \text{in } \Omega, \\ { \mathcal{N}_{s}u=0 } & \text{in } \mathbb{R}^{n}\setminus \overline{\Omega}, \end{cases} \end{align*} $$

where Abstract Image$\Omega \subset \mathbb {R}^{n}$ is a bounded domain of class Abstract Image$C^{1,1}$, Abstract Image$1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \}, 0<s<1, d>0$ and Abstract Image$\mathcal {N}_{s}u$ is the nonlocal Neumann derivative. We show that for small Abstract Image$d,$ the least energy solutions Abstract Image$u_d$ of the above problem achieve an Abstract Image$L^{\infty }$-bound independent of Abstract Image$d.$ Using this together with suitable Abstract Image$L^{r}$-estimates on Abstract Image$u_d,$ we show that the least energy solution Abstract Image$u_d$ achieves a maximum on the boundary of Abstract Image$\Omega $ for d sufficiently small.

分数新曼问题最小能量解的渐近行为
我们研究了以下一类非局部新曼问题的最小能量解的渐近行为: $$ \begin{align*}\开始{ d(-\Delta)^{s}u+ u= \vert u\vert^{p-1}u } & (text{in }\Omega, \{u>0} & \text{in }\Omega, \ { \mathcal{N}_{s}u=0 } & \text{in }\mathbb{R}^{n}\setminus \overline{Omega}, \end{cases}\end{align*}其中 $\Omega \subset \mathbb {R}^{n}$ 是类 $C^{1,1}$ 的有界域,$1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \},0<s<1,d>0$,$\mathcal {N}_{s}u$ 是非局部诺依曼导数。我们证明,对于较小的 $d$,上述问题的最小能量解 $u_d$ 实现了与 $d 无关的 $L^{/infty }$ 约束。$ 利用这一点以及对 $u_d$ 的适当 $L^{r}$ 估计,我们证明,对于足够小的 d,最小能量解 $u_d$ 在 $\Omega $ 的边界上实现了最大值。
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来源期刊
CiteScore
1.70
自引率
0.00%
发文量
36
审稿时长
6 months
期刊介绍: The Journal of the Australian Mathematical Society is the oldest journal of the Society, and is well established in its coverage of all areas of pure mathematics and mathematical statistics. It seeks to publish original high-quality articles of moderate length that will attract wide interest. Papers are carefully reviewed, and those with good introductions explaining the meaning and value of the results are preferred. Published Bi-monthly Published for the Australian Mathematical Society
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