{"title":"On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities","authors":"Qiu Xiang Bian, Jing Chen, Jing Yang","doi":"10.1007/s10114-023-2244-1","DOIUrl":null,"url":null,"abstract":"<div><p>This paper deals with the following prescribed boundary mean curvature problem in <span>\\({\\mathbb{B}^N}\\)</span></p><div><div><span>$$\\left\\{ {\\matrix{{ - \\Delta u = 0,\\,u > 0,} \\hfill & {y \\in {\\mathbb{B}^N},} \\hfill \\cr {{{\\partial u} \\over {\\partial \\nu }} + {{N - 2} \\over 2}u = {{N - 2} \\over 2}\\tilde K(y){u^{{2^\\sharp } - 1}},} \\hfill & {y \\in {\\mathbb{S}^{N - 1}},} \\hfill \\cr } } \\right.$$</span></div></div><p>where <span>\\(\\tilde K(y) = \\tilde K(|{y^\\prime }|,\\tilde y)\\)</span> is a bounded nonnegative function with <span>\\(y = ({y^\\prime },\\tilde y) \\in {\\mathbb{R}^2} \\times {\\mathbb{R}^{N - 3}},\\,\\,{2^\\sharp } = {{2(N - 1)} \\over {N - 2}}\\)</span>. Combining the finite-dimensional reduction method and local Pohozaev type of identities, we prove that if <i>N</i> ≥ 5 and <span>\\(\\tilde K(r,\\tilde y)\\)</span> has a stable critical point (<i>r</i><sub>0</sub>, <span>\\(({r_0},{\\tilde y_0})\\)</span>) with <i>r</i><sub>0</sub> > 0 and <span>\\(\\tilde K({r_0},{\\tilde y_0}) > 0\\)</span>, then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here our result fill the gap that the above critical points may include the saddle points of <span>\\(\\tilde K(r,\\tilde y)\\)</span>.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-2244-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper deals with the following prescribed boundary mean curvature problem in \({\mathbb{B}^N}\)
where \(\tilde K(y) = \tilde K(|{y^\prime }|,\tilde y)\) is a bounded nonnegative function with \(y = ({y^\prime },\tilde y) \in {\mathbb{R}^2} \times {\mathbb{R}^{N - 3}},\,\,{2^\sharp } = {{2(N - 1)} \over {N - 2}}\). Combining the finite-dimensional reduction method and local Pohozaev type of identities, we prove that if N ≥ 5 and \(\tilde K(r,\tilde y)\) has a stable critical point (r0, \(({r_0},{\tilde y_0})\)) with r0 > 0 and \(\tilde K({r_0},{\tilde y_0}) > 0\), then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here our result fill the gap that the above critical points may include the saddle points of \(\tilde K(r,\tilde y)\).
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.