Positivity

, Volume 12, Issue 3, pp 555–569

Positive solutions for second-order superlinear repulsive singular Neumann boundary value problems

  • Jifeng Chu
  • Xiaoning Lin
  • Daqing Jiang
  • Donal O’Regan
  • Ravi P. Agarwal
Article

Abstract

In this paper we establish the multiplicity of positive solutions to second-order superlinear repulsive singular Neumann boundary value problems. It is proved that such a problem has at least two positive solutions under reasonable conditions. Our nonlinearity may be repulsive singular in its dependent variable and superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.

Mathematics Subject Classification (2000)

34B15 

Keywords

Superlinear repulsive singular Neumann boundary value problems positive solutions leray-Schauder alternative fixed point theorem in cones 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Cabada, L. Sanchez, A positive operator approach to the Neumann problem for a second order ordinary differential equation, J. Math. Anal. Appl., 204 (1996), 774–785.Google Scholar
  2. 2.
    A. Cabada, R.L. Pouso, Existence result for the problem (ϕ(u′))′ = f(t, u, u′) with periodic and Neumann boundary conditions, Nonlinear Anal., 30 (1997), 1733–1742.Google Scholar
  3. 3.
    A. Cabada, P. Habets, R.L. Pouso, Optimal existence conditions for ϕ-Laplacian equations with upper and lower solutions in the reversed order, J. Differ. Equ., 166 (2000), 385–401.Google Scholar
  4. 4.
    A. Cabada, P. Habets, S. Lois, Monotone method for the Neumann problem with lower and upper solutions in the reverse order, Appl. Math. Comput., 117 (2001), 1–14.Google Scholar
  5. 5.
    M. Cherpion, C. De Coster, P. Habets, A constructive monotone iterative method for second order BVP in the presence of lower and upper solutions, Appl. Math. Comput., 123 (2001), 75–91.Google Scholar
  6. 6.
    J. Chu, Y. Sun, H. Chen, Positive solutions of Neumann problems with singularities, J. Math. Anal. Appl., 337 (2008), 1267–1272.Google Scholar
  7. 7.
    H. Dang, S.F. Oppenheimer, Existence and uniqueness results for some nonlinear boundary value problems, J. Math. Anal. Appl., 198 (1996), 35–48.Google Scholar
  8. 8.
    K. Deimling, Nonlinear Functional Analysis, Springer, New York (1985).Google Scholar
  9. 9.
    Y. Dong, A Neumann problem at resonance with the nonlinearity restricted in one direction, Nonlinear Anal., 51 (2002), 739–747.Google Scholar
  10. 10.
    L.H. Erbe, H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Am. Math. Soc., 120 (1994), 743–748.Google Scholar
  11. 11.
    D. Jiang, H. Liu, Existence of positive solutions to second order Neumann boundary value problem, J. Math. Res. Expos., 20 (2000), 360–364.Google Scholar
  12. 12.
    D. Jiang, J. Chu, D. O’Regan, R.P. Agarwal, Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, J. Math. Anal. Appl., 286 (2003), 563–576.Google Scholar
  13. 13.
    R. Ma, Existence of positive radial solutions for elliptic systems, J. Math. Anal. Appl., 201 (1996), 375–386.Google Scholar
  14. 14.
    I. Rachůnková, S. Staňek, Topological degree method in functional boundary value problems at resonance, Nonlinear Anal., 27 (1996), 271–285.Google Scholar
  15. 15.
    I. Rachůnková, Upper and lower solutions with inverse inequality, Ann. Polon. Math., 65 (1997), 235–244.Google Scholar
  16. 16.
    J. Sun, W. Li, Multiple positive solutions to second order Neumann boundary value problems, Appl. Math. Comput., 146 (2003), 187–194.Google Scholar
  17. 17.
    N. Yazidi, Monotone method for singular Neumann problem, Nonlinear Anal., 49 (2002), 589–602.Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  • Jifeng Chu
    • 1
  • Xiaoning Lin
    • 2
  • Daqing Jiang
    • 2
  • Donal O’Regan
    • 3
  • Ravi P. Agarwal
    • 4
  1. 1.College of ScienceHohai UniversityNanjingChina
  2. 2.School of Mathematics and StatisticsNortheast Normal UniversityChangchunChina
  3. 3.Department of MathematicsNational University of IrelandGalwayIreland
  4. 4.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA

Personalised recommendations