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Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

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Abstract

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.

Mathematics Subject Classification (2000)

Primary: 35P15 35J20 Secondary: 92D25 

Keywords

Semilinear elliptic problem Indefinite weight Positive solution Bifurcation Global subcontinuum A priori bound Multiplicity Population dynamics 
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References

  1. 1.
    Afrouzi G.A., Brown K.J.: On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions. Proc. Am. Math. Soc. 127, 125–130 (1999)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Amann, H.: Nonlinear elliptic equations with nonlinear boundary conditions. New developments in differential equations. In: Proceedings of the 2nd Scheveningen Conference, Scheveningen, 1975. North-Holland Mathematical Studies, vol. 21, pp. 43–63. North-Holland, Amsterdam (1976)Google Scholar
  3. 3.
    Brown K.J.: Local and global bifurcation results for a semilinear boundary value problem. J. Differ. Equ. 239, 296–310 (2007)MATHCrossRefGoogle Scholar
  4. 4.
    Brown K.J., Lin S.S.: On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function. J. Math. Anal. Appl. 75, 112–120 (1980)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cano-Casanova S.: Compact components of positive solutions for superlinear indefinite elliptic problems of mixed type. Topol. Methods Nonlinear Anal. 23, 45–72 (2004)MATHMathSciNetGoogle Scholar
  6. 6.
    Cantrell R.S., Cosner C.: Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. Wiley, Chichester (2003)Google Scholar
  7. 7.
    Cantrell R.S., Cosner C., Martinez S.: Global bifurcation of solutions to diffusive logistic equations on bounded domains subject to nonlinear boundary conditions. Proc. R. Soc. Edinb. 139A, 45–56 (2009)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Crandall M.G., Rabinowitz P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dancer E.N.: Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one. Bull. Lond. Math. Soc. 34, 533–538 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Filo J., Kačur J.: Local existence of general nonlinear parabolic systems. Nonlinear Anal. TMA 24, 1597–1618 (1995)MATHCrossRefGoogle Scholar
  11. 11.
    García-Melián J., Morales-Rodrigo C., Rossi J.D., Suárez A.: Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary. Ann. Mat. Pura Appl. 187, 459–486 (2008)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 editionGoogle Scholar
  13. 13.
    López-Gómez J.: Spectral Theory and Nonlinear Functional Analysis. Research Notes in Mathematics 426. Chapman & Hall/CRC, Boca Raton (2001)Google Scholar
  14. 14.
    López-Gómez J., Molina-Meyer M.: Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas. J. Differ. Equ. 209, 416–441 (2005)MATHCrossRefGoogle Scholar
  15. 15.
    Morales-Rodrigo C., Suárez A.: Some elliptic problems with nonlinear boundary conditions. In: Cano-Casanova, S., López-Gómez, J., Mora-Corral, C. (eds) Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, pp. 175–199. World Scientific Publishing Company, Hackensack (2005)CrossRefGoogle Scholar
  16. 16.
    Protter M.H., Weinberger H.F.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs (1967)Google Scholar
  17. 17.
    Rossi, J.D.: Elliptic problems with nonlinear boundary conditions and the Sobolev trace theorem. Stationary Partial Differential Equations, vol. II, pp. 311–406, Handbook of the Differential Equations. Elsevier/North-Holland, Amsterdam (2005)Google Scholar
  18. 18.
    Umezu, K.: One parameter-dependent nonlinear elliptic boundary value problems arising in population dynamics. Advances in Analysis, pp. 177–186. World Scientific Publishing Company, Hackensack (2005)Google Scholar
  19. 19.
    Umezu K.: On eigenvalue problems with Robin type boundary conditions having indefinite coefficients. Appl. Anal. 85, 1313–1325 (2006)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Umezu K.: Positive solutions of semilinear elliptic eigenvalue problems with concave nonlinearities. Adv. Differ. Equ. 12, 1415–1436 (2007)MATHMathSciNetGoogle Scholar

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© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationIbaraki UniversityMitoJapan

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