{"title":"Radial Positive Solutions for Semilinear Elliptic Problems with Linear Gradient Term in $$\\mathbb {R}^N$$","authors":"Ruyun Ma, Xiaoxiao Su, Zhongzi Zhao","doi":"10.1007/s12220-024-01787-3","DOIUrl":null,"url":null,"abstract":"<p>We are concerned with the linear problem </p><span>$$\\begin{aligned} \\left\\{ \\begin{array}{ll} -\\Delta u+\\frac{\\kappa }{|x|^2} x\\cdot \\nabla u =\\lambda K(|x|) u, & x\\in \\mathbb {R}^N,\\\\ u(x)>0, & x\\in \\mathbb {R}^N,\\\\[2ex] u(x)\\rightarrow 0, & |x|\\rightarrow \\infty , \\end{array} \\right. \\end{aligned}$$</span><p>where <span>\\(\\lambda \\)</span> is a positive parameter, <span>\\(\\kappa \\in [0,N-2)\\)</span>, <span>\\(N> 2\\)</span> and <span>\\(K:\\mathbb {R}^N \\rightarrow (0,\\infty )\\)</span> is continuous and satisfies certain decay assumptions. We obtain the existence of the principal eigenvalue <span>\\(\\lambda _1^{\\text {rad}}\\)</span> and the corresponding positive eigenfunction <span>\\(\\varphi _1\\)</span> satisfies <span>\\(\\lim \\nolimits _{|x|\\rightarrow \\infty }\\varphi _1(|x|)=\\frac{c}{|x|^{N-2-\\kappa }}\\)</span> for some <span>\\(c>0\\)</span>. As applications, we also study the existence of connected component of positive solutions for nonlinear infinite semipositone elliptic problems by bifurcation techniques.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"214 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Geometric Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s12220-024-01787-3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
where \(\lambda \) is a positive parameter, \(\kappa \in [0,N-2)\), \(N> 2\) and \(K:\mathbb {R}^N \rightarrow (0,\infty )\) is continuous and satisfies certain decay assumptions. We obtain the existence of the principal eigenvalue \(\lambda _1^{\text {rad}}\) and the corresponding positive eigenfunction \(\varphi _1\) satisfies \(\lim \nolimits _{|x|\rightarrow \infty }\varphi _1(|x|)=\frac{c}{|x|^{N-2-\kappa }}\) for some \(c>0\). As applications, we also study the existence of connected component of positive solutions for nonlinear infinite semipositone elliptic problems by bifurcation techniques.