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Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity

  • Uriel Kaufmann
  • Humberto Ramos Quoirin
  • Kenichiro Umezu
Article

Abstract

Let \(\Omega \subset \mathbb {R}^{N}\) (\(N\ge 1\)) be a bounded and smooth domain and \(a:\Omega \rightarrow \mathbb {R}\) be a sign-changing weight satisfying \(\int _{\Omega }a<0\). We prove the existence of a positive solution \(u_{q}\) for the problem if \(q_{0}<q<1\), for some \(q_{0}=q_{0}(a)>0\). In doing so, we improve the existence result previously established in Kaufmann et al. (J Differ Equ 263:4481–4502, 2017). In addition, we provide the asymptotic behavior of \(u_{q}\) as \(q\rightarrow 1^{-}\). When \(\Omega \) is a ball and a is radial, we give some explicit conditions on q and a ensuring the existence of a positive solution of \((P_{a,q})\). We also obtain some properties of the set of q’s such that \((P_{a,q})\) admits a solution which is positive on \(\overline{\Omega }\). Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the sub-supersolution method.

Keywords

Elliptic problem Indefinite Sublinear Positive solution 

Mathematics Subject Classification

35J25 35J61 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Uriel Kaufmann
    • 1
  • Humberto Ramos Quoirin
    • 2
  • Kenichiro Umezu
    • 3
  1. 1.FaMAFUniversidad Nacional de CórdobaCórdobaArgentina
  2. 2.Universidad de Santiago de ChileSantiagoChile
  3. 3.Department of Mathematics, Faculty of EducationIbaraki UniversityMitoJapan

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