{"title":"有低阶项的非线性椭圆方程的诺依曼问题","authors":"","doi":"10.1016/j.na.2024.113626","DOIUrl":null,"url":null,"abstract":"<div><p>In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><mi>λ</mi><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>u</mi><mo>−</mo><mo>div</mo><mrow><mo>(</mo><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>=</mo><mi>f</mi><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow></mfenced><mi>⋅</mi><munder><mrow><mi>n</mi></mrow><mo>̲</mo></munder><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mtext>on</mtext><mspace></mspace><mi>∂</mi><mi>Ω</mi><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mi>Ω</mi></math></span> is a bounded domain of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, with Lipschitz boundary, <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></math></span> , <span><math><munder><mrow><mi>n</mi></mrow><mo>̲</mo></munder></math></span> is the outer unit normal to <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>, <span><math><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></math></span>, the datum <span><math><mi>f</mi></math></span> belongs to the dual space of <span><math><mrow><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> or to Lebesgue space <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>. Finally the coefficients <span><math><mi>b</mi></math></span>, <span><math><mi>c</mi></math></span> belong to appropriate Lebesgue spaces or Lorentz spaces.</p><p>Existence results for weak solutions or renormalized solutions are proved under smallness assumptions on the coefficients <span><math><mi>b</mi></math></span> and <span><math><mi>c</mi></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001457/pdfft?md5=ce03c34a8fe445e869b1bd2082487f52&pid=1-s2.0-S0362546X24001457-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Neumann problems for nonlinear elliptic equations with lower order terms\",\"authors\":\"\",\"doi\":\"10.1016/j.na.2024.113626\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is <span><span><span><math><mfenced><mrow><mtable><mtr><mtd><mi>λ</mi><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>u</mi><mo>−</mo><mo>div</mo><mrow><mo>(</mo><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>=</mo><mi>f</mi><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow></mfenced><mi>⋅</mi><munder><mrow><mi>n</mi></mrow><mo>̲</mo></munder><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mtext>on</mtext><mspace></mspace><mi>∂</mi><mi>Ω</mi><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>where <span><math><mi>Ω</mi></math></span> is a bounded domain of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, with Lipschitz boundary, <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></math></span> , <span><math><munder><mrow><mi>n</mi></mrow><mo>̲</mo></munder></math></span> is the outer unit normal to <span><math><mrow><mi>∂</mi><mi>Ω</mi></mrow></math></span>, <span><math><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></math></span>, the datum <span><math><mi>f</mi></math></span> belongs to the dual space of <span><math><mrow><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> or to Lebesgue space <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>. Finally the coefficients <span><math><mi>b</mi></math></span>, <span><math><mi>c</mi></math></span> belong to appropriate Lebesgue spaces or Lorentz spaces.</p><p>Existence results for weak solutions or renormalized solutions are proved under smallness assumptions on the coefficients <span><math><mi>b</mi></math></span> and <span><math><mi>c</mi></math></span>.</p></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-08-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001457/pdfft?md5=ce03c34a8fe445e869b1bd2082487f52&pid=1-s2.0-S0362546X24001457-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/S0362546X24001457\",\"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/S0362546X24001457","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
本文证明了原型为 λ|u|p-2u-Δpu-div(c(x)|u|p-2u)+b(x)|∇u|p-2∇u=finΩ 的非线性椭圆 Neumann 问题解的存在性结果、|∇u|p-2∇u+c(x)|u|p-2u⋅n̲=0∂Ω,其中 Ω 是 RN 的有界域,N≥2,具有 Lipschitz 边界,1<;p<N ,n̲是∂Ω的外单位法线,λ>0,基准 f 属于 W1,p(Ω) 的对偶空间或 Lebesgue 空间 L1(Ω)。最后,系数 b、c 属于适当的 Lebesgue 空间或洛伦兹空间。在系数 b 和 c 的小性假设下,证明了弱解或重规范化解的存在性结果。
Neumann problems for nonlinear elliptic equations with lower order terms
In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is where is a bounded domain of , , with Lipschitz boundary, , is the outer unit normal to , , the datum belongs to the dual space of or to Lebesgue space . Finally the coefficients , belong to appropriate Lebesgue spaces or Lorentz spaces.
Existence results for weak solutions or renormalized solutions are proved under smallness assumptions on the coefficients and .
期刊介绍:
Nonlinear Analysis focuses on papers that address significant problems in Nonlinear Analysis that have a sustainable and important impact on the development of new directions in the theory as well as potential applications. Review articles on important topics in Nonlinear Analysis are welcome as well. In particular, only papers within the areas of specialization of the Editorial Board Members will be considered. Authors are encouraged to check the areas of expertise of the Editorial Board in order to decide whether or not their papers are appropriate for this journal. The journal aims to apply very high standards in accepting papers for publication.