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


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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  1. [1]
    S. Alama, Semilinear elliptic equations with sublinear indefinite nonlinearities, Advances in Differential Equations 4 (1999), 813–842.MathSciNetMATHGoogle Scholar
  2. [2]
    H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, Journal of Differential Equations 146 (1998), 336–374.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, Journal Functional Analysis 122 (1994), 519–543.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    C. Bandle, A. M. Pozio and A. Tesei, Existence and uniqueness of solutions of nonlinear Neumann problems, Mathematische Zeitschrift 199 (1988), 257–278.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proceedings of the Royal Society of Edinburgh 97A (1984), 21–34.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    K. J. Brown, Local and global bifurcation results for a semilinear boundary value problem, Journal of Differential Equations 239 (2007), 296–310.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, Journal of Differential Equations 193 (2003), 481–499.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    S. Cano-Casanova, Nonlinear mixed boundary conditions in BVPs of logistic type with spatial heterogeneities and a nonlinear flux on the boundary with arbitrary sign. The case p > 2q - 1, Journal of Differential Equations 256 (2014), 82–107.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proceedings of the Royal Society of Edinburgh 112A (1989), 293–318.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal fo Functional Analysis 8 (1971), 321–340.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    M. Delgado and A. Suárez, Positive solutions for the degenerate logistic indefinite superlinear problem: the slow diffusion case, Houston Journal of Mathematics 29 (2003), 801–820.MathSciNetMATHGoogle Scholar
  12. [12]
    D. G. De Figueiredo, J.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, Journal of Functional Analysis 199 (2003), 452–467.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    D. G. de Figueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, Journal of the European Mathematical Society 8 (2006), 269–286.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    J. Garcia-Azorero, I. Peral, and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition, Journal of Differential Equations 198 (2004), 91–128.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    P. Hess, On the solvability of nonlinear elliptic boundary value problems, Indiana University Mathematics Journal 25 (1976), 461–466.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Communications in Partial Differential Equations 10 (1980), 999–1030.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    P. Korman, On uniqueness of positive solutions for a class of semilinear equations, Discrete and Continuous Dynamical Systems 8 (2002), 865–871.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    P. Korman, Exact multiplicity and numerical computation of solutions for two classes of non-autonomous problems with concave-convex nonlinearities, Nonlinear Analysis 93 (2013), 226–235.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall/CRC Research Notes in Mathematics, Vol. 426, Chapman & Hall/CRC, Boca Raton, FL, 2001.CrossRefMATHGoogle Scholar
  20. [20]
    J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing, Hackensack, NJ, 2013.CrossRefMATHGoogle Scholar
  21. [21]
    J. López-Gómez and M. Molina-Meyer, Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas, Journal of Differential Equations 209 (2005), 416–441.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    J. López-Gómez, M. Molina-Meyer and A. Tellini, The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions, Journal of Differential Equations 255 (2013), 503–523.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    F. O. de Paiva, Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity, Journal of Functional Analysis 261 (2011), 2569–2586.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis 7 (1971), 487–513.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    H. Ramos Quoirin and K. Umezu, Bifurcation for a logistic elliptic equation with nonlinear boundary conditions: A limiting case, Journal of Mathematical Analysis and Applications 428 (2015), 1265–1285.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    H. Ramos Quoirin and K. Umezu, A loop type component in the non-negative solutions set of an indefinite problem, preprint, arXiv:1610.00964.Google Scholar
  27. [27]
    N. Tarfulea, Existence of positive solutions of some nonlinear Neumann problems, Analele Universităţii din Craiova. Seria Matematică-Informatică 23 (1996), 9–18.MathSciNetMATHGoogle Scholar
  28. [28]
    K. Umezu, Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions, NoDEA Nonlinear Differential Equations and Applications 17 (2010), 323–336.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    G. T. Whyburn, Topological Analysis, Princeton Mathematical Series, Vol. 23, Princeton University Press, Princeton, NJ, 1964.CrossRefMATHGoogle Scholar
  30. [30]
    T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, Journal of Mathematical Analysis and Applications 318 (2006), 253–270.MathSciNetCrossRefMATHGoogle Scholar

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© 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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