{"title":"Infinitely many positive energy solutions for semilinear Neumann equations with critical Sobolev exponent and concave-convex nonlinearity","authors":"Rachid Echarghaoui, Rachid Sersif, Zakaria Zaimi","doi":"10.1007/s13348-023-00426-4","DOIUrl":null,"url":null,"abstract":"<p>The authors of Cao and Yan (J Differ Equ 251:1389–1414, 2011) have considered the following semilinear critical Neumann problem </p><span>$$\\begin{aligned} \\varvec{-\\Delta u=\\vert u\\vert ^{2^{*}-2} u+g(u) \\quad \\text{ in } \\Omega , \\quad \\frac{\\partial u}{\\partial \\nu }=0 \\quad \\text{ on } \\partial \\Omega ,} \\end{aligned}$$</span><p>where <span>\\(\\varvec{\\Omega }\\)</span> is a bounded domain in <span>\\(\\varvec{\\mathbb {R}^{N}}\\)</span> satisfying some geometric conditions, <span>\\(\\varvec{\\nu }\\)</span> is the outward unit normal of <span>\\(\\varvec{\\partial \\Omega , 2^{*}:=\\frac{2 N}{N-2}}\\)</span> and <span>\\(\\varvec{g(t):=\\mu \\vert t\\vert ^{p-2} t-t,}\\)</span> where <span>\\(\\varvec{p \\in \\left( 2,2^{*}\\right) }\\)</span> and <span>\\(\\varvec{\\mu >0}\\)</span> are constants. They proved the existence of infinitely many solutions with positive energy for the above problem if <span>\\(\\varvec{N>\\max \\left( \\frac{2(p+1)}{p-1}, 4\\right) .}\\)</span> In this present paper, we consider the case where the exponent <span>\\(\\varvec{p \\in \\left( 1,2\\right) }\\)</span> and we show that if <span>\\(\\varvec{N>\\frac{2(p+1)}{p-1},}\\)</span> then the above problem admits an infinite set of solutions with positive energy. Our main result extend that obtained by P. Han in [9] for the case of elliptic problem with Dirichlet boundary conditions.</p>","PeriodicalId":50993,"journal":{"name":"Collectanea Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Collectanea Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s13348-023-00426-4","RegionNum":2,"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
The authors of Cao and Yan (J Differ Equ 251:1389–1414, 2011) have considered the following semilinear critical Neumann problem
where \(\varvec{\Omega }\) is a bounded domain in \(\varvec{\mathbb {R}^{N}}\) satisfying some geometric conditions, \(\varvec{\nu }\) is the outward unit normal of \(\varvec{\partial \Omega , 2^{*}:=\frac{2 N}{N-2}}\) and \(\varvec{g(t):=\mu \vert t\vert ^{p-2} t-t,}\) where \(\varvec{p \in \left( 2,2^{*}\right) }\) and \(\varvec{\mu >0}\) are constants. They proved the existence of infinitely many solutions with positive energy for the above problem if \(\varvec{N>\max \left( \frac{2(p+1)}{p-1}, 4\right) .}\) In this present paper, we consider the case where the exponent \(\varvec{p \in \left( 1,2\right) }\) and we show that if \(\varvec{N>\frac{2(p+1)}{p-1},}\) then the above problem admits an infinite set of solutions with positive energy. Our main result extend that obtained by P. Han in [9] for the case of elliptic problem with Dirichlet boundary conditions.
期刊介绍:
Collectanea Mathematica publishes original research peer reviewed papers of high quality in all fields of pure and applied mathematics. It is an international journal of the University of Barcelona and the oldest mathematical journal in Spain. It was founded in 1948 by José M. Orts. Previously self-published by the Institut de Matemàtica (IMUB) of the Universitat de Barcelona, as of 2011 it is published by Springer.