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Israel Journal of Mathematics

, Volume 220, Issue 1, pp 103–160 | Cite as

An indefinite concave-convex equation under a Neumann boundary condition I

  • Humberto Ramos Quoirin
  • Kenichiro Umezu
Article

Abstract

We investigate the problem (P λ) −Δu = λb(x)|u| q−2 u + a(x)|u| p−2 u in Ω, ∂u/n = 0 on Ω, where Ω is a bounded smooth domain in R N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b\({C^\alpha }\left( {\overline \Omega } \right)\) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P λ).

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Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversidad de Santiago de ChileSantiagoChile
  2. 2.Department of Mathematics, Faculty of EducationIbaraki UniversityMitoJapan

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