{"title":"A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem","authors":"Yong-Hoon Lee, Rui Yang","doi":"10.1515/ans-2023-0107","DOIUrl":null,"url":null,"abstract":"Abstract In this article, we study the uniqueness of positive symmetric solutions of the following mean curvature problem in Euclidean space: (P) <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mfenced open=\"{\" close=\"\"> <m:mrow> <m:mtable displaystyle=\"true\"> <m:mtr> <m:mtd columnalign=\"left\"> <m:msup> <m:mrow> <m:mfenced open=\"(\" close=\")\"> <m:mrow> <m:mfrac> <m:mrow> <m:mi>u</m:mi> <m:mo accent=\"false\">′</m:mo> </m:mrow> <m:mrow> <m:msqrt> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo accent=\"false\">′</m:mo> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msqrt> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mo accent=\"true\">′</m:mo> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>h</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>x</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mspace width=\"1.0em\" /> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width=\"1.0em\" /> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \\left\\{\\begin{array}{l}{\\left(\\frac{u^{\\prime} }{\\sqrt{1+{| u^{\\prime} | }^{2}}}\\right)}^{^{\\prime} }+h\\left(x)f\\left(u)=0,\\hspace{1em}-1\\lt x\\lt 1,\\hspace{1.0em}\\\\ u\\left(-1)=u\\left(1)=0,\\hspace{1.0em}\\end{array}\\right. where <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>h</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> h\\in {C}^{1}\\left(\\left[-1,1]) and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>;</m:mo> <m:mspace width=\"0.33em\" /> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> f\\in {C}^{1}\\left(\\left[0,\\infty );\\hspace{0.33em}\\left[0,\\infty )) . Under suitable conditions on <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>h</m:mi> </m:math> h and monotone condition on <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mfrac> <m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:mfrac> </m:math> \\frac{f\\left(s)}{s} , by introducing a modified Picone-type identity, we prove that the problem has at most one positive symmetric solution.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"20 1","pages":"0"},"PeriodicalIF":2.1000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advanced Nonlinear Studies","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/ans-2023-0107","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
Abstract In this article, we study the uniqueness of positive symmetric solutions of the following mean curvature problem in Euclidean space: (P) u′1+∣u′∣2′+h(x)f(u)=0,−1<x<1,u(−1)=u(1)=0, \left\{\begin{array}{l}{\left(\frac{u^{\prime} }{\sqrt{1+{| u^{\prime} | }^{2}}}\right)}^{^{\prime} }+h\left(x)f\left(u)=0,\hspace{1em}-1\lt x\lt 1,\hspace{1.0em}\\ u\left(-1)=u\left(1)=0,\hspace{1.0em}\end{array}\right. where h∈C1([−1,1]) h\in {C}^{1}\left(\left[-1,1]) and f∈C1([0,∞);[0,∞)) f\in {C}^{1}\left(\left[0,\infty );\hspace{0.33em}\left[0,\infty )) . Under suitable conditions on h h and monotone condition on f(s)s \frac{f\left(s)}{s} , by introducing a modified Picone-type identity, we prove that the problem has at most one positive symmetric solution.
期刊介绍:
Advanced Nonlinear Studies is aimed at publishing papers on nonlinear problems, particulalry those involving Differential Equations, Dynamical Systems, and related areas. It will also publish novel and interesting applications of these areas to problems in engineering and the sciences. Papers submitted to this journal must contain original, timely, and significant results. Articles will generally, but not always, be published in the order when the final copies were received.