Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem

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• 作者：Alberto Boscaggin (1)
  Fabio Zanolin (2)
  
  1. Department of Mathematics ; University of Torino ; Via Carlo Alberto 10 ; 10123聽 ; Torino ; Italy
  2. Department of Mathematics and Computer Science ; University of Udine ; Via delle Scienze 206 ; 33100聽 ; Udine ; Italy
  
• 关键词：Boundary value problems ; Indefinite weight ; Necessary and sufficient solvability conditions ; 34B15 ; 34B09
• 刊名：Annali di Matematica Pura ed Applicata
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：194
• 期：2
• 页码：451-478
• 全文大小：518 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1618-1891

文摘

We study the second-order nonlinear differential equation \(u'' + a(t) g(u) = 0\) , where \(g\) is a continuously differentiable function of constant sign defined on an open interval \(I\subseteq {\mathbb R}\) and \(a(t)\) is a sign-changing weight function. We look for solutions \(u(t)\) of the differential equation such that \(u(t)\in I,\) satisfying the Neumann boundary conditions. Special examples, considered in our model, are the equations with singularity, for \(I = {\mathbb R}^+_0\) and \(g(u) \sim - u^{-\sigma },\) as well as the case of exponential nonlinearities, for \(I = {\mathbb R}\) and \(g(u) \sim \exp (u)\) . The proofs are obtained by passing to an equivalent equation of the form \(x'' = f(x)(x')^2 + a(t)\) .