Nodal solutions of weighted indefinite problems

Abstract

This paper analyzes the structure of the set of nodal solutions, i.e., solutions changing sign, of a class of one-dimensional superlinear indefinite boundary value problems with indefinite weight functions in front of the spectral parameter. Quite surprisingly, the associated high-order eigenvalues may not be concave as is the case for the lowest one. As a consequence, in many circumstances, the nodal solutions can bifurcate from three or even four bifurcation points from the trivial solution. This paper combines analytical and numerical tools. The analysis carried out is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously the numerical study confirms and illuminates the analysis.

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References

  1. 1.

    E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, SIAM Classics in Applied Mathematics 45, SIAM, Philadelphia, 2003.

  2. 2.

    I. Antón and J. López-Gómez, Principal eigenvalues of weighted periodic-parabolic problems, Rend. Istit. Mat. Univ. Trieste 49 (2017), 287–318.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics vol. 96, Birkhäuser/Springer, Basel, 2011.

    Google Scholar 

  4. 4.

    W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One Parameter Semigroups of Positive Operators, Lectures Notes in Mathematics 1184, Berlin, Springer, 1986.

  5. 5.

    H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operator sin general domains, Comm. Pure Appl. Math. 47 (1994), 47–92.

    MathSciNet  Article  Google Scholar 

  6. 6.

    F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, part I: Branches of nonsingular solutions, Numer. Math. 36 (1980), 1–25.

    MathSciNet  Article  Google Scholar 

  7. 7.

    F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, part II: Limit points, Numer. Math. 37 (1981), 1–28.

    MathSciNet  Article  Google Scholar 

  8. 8.

    F. Brezzi, J. Rappaz and P. A. Raviart, Finite dimensional approximation of nonlinear problems, part III: Simple bifurcation points, Numer. Math. 38 (1981), 1–30.

    MathSciNet  Article  Google Scholar 

  9. 9.

    G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems, Clarendon Press, Oxford, 1998.

    Google Scholar 

  10. 10.

    S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of nonclassical mixed boundary value problems, J. Dif. Eqns. 178 (2002), 123–211.

    Article  Google Scholar 

  11. 11.

    C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Mechanics, Springer, Berlin, Germany, 1988.

    Google Scholar 

  12. 12.

    M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340.

    MathSciNet  Article  Google Scholar 

  13. 13.

    M. Crouzeix and J. Rappaz, On Numerical Approximation in Bifurcation Theory, Recherches en Mathématiques Appliquées 13, Masson, Paris, 1990.

  14. 14.

    E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and bifurcation software for ODEs, 2012, http://www.dam.brown.edu/people/sandsted/auto/auto07p.pdf.

  15. 15.

    J. Esquinas and J. López-Gómez, Optimal multiplicity in local bifurcation theory: Generalized generic eigenvalues, J. Diff. Eqns. 71 (1988), 72–92.

    MathSciNet  Article  Google Scholar 

  16. 16.

    J. C. Eilbeck, The pseudo-spectral method and path-following in reaction-diffusion bifurcation studies, SIAM J. Sci. Stat. Comput. 7 (1986), 599–610.

    MathSciNet  Article  Google Scholar 

  17. 17.

    D. G. de Figueiredo, Positive Solutions of Semilinear Elliptic Problems, Lectures Notes of a Latin-American School on Differential Equations, Sao Paolo 1981, Lectures Notes in Mathematics 957 (pp. 34–87), Springer, 1982.

  18. 18.

    R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction diffusion equations, J. Diff. Eqns. 167 (2000), 36–72.

    MathSciNet  Article  Google Scholar 

  19. 19.

    R. Gómez-Reñasco and J. López-Gómez, On the existence and numerical computation of classical and non-classical solutions for a family of elliptic boundary value problems, Nonl. Anal. TMA 48 (2002), 567–605.

    MathSciNet  Article  Google Scholar 

  20. 20.

    P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Part. Dif. Eqns. 5 (1980), 999–1030.

    MathSciNet  Article  Google Scholar 

  21. 21.

    T. Kato, Superconvexity of the spectral radius and convexity of the spectral bound and the type, Math. Z. 180 (1982), 265–273.

    MathSciNet  Article  Google Scholar 

  22. 22.

    T. Kato, Perturbation Theory for Linear Operators, Springer, 1995.

  23. 23.

    H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Tata Insitute of Fundamental Research, Springer, Berlin, Germany, 1986.

    Google Scholar 

  24. 24.

    H. B. Keller and Z. H. Yang, A direct method for computing higher order folds, SIAM J. Sci. Stat. 7 (1986), 351–361.

    MathSciNet  MATH  Google Scholar 

  25. 25.

    J. López-Gómez Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéericos, Cuadernos de Matemática y Mecánica, Serie Cursos y Seminarios 4, Santa Fe, R. Argentina, 1988.

  26. 26.

    J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, CRC Press, Boca Raton, 2001.

    Google Scholar 

  27. 27.

    J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing, 2013.

  28. 28.

    J. López-Gómez, J. C. Eilbeck, K. Duncan and M. Molina-Meyer, Structure of solution manifolds in a strongly coupled elliptic system, IMA J. Numer. Anal. 12 (1992), 405–428.

    MathSciNet  Article  Google Scholar 

  29. 29.

    J. López-Gómez and M. Molina-Meyer, Bounded components of positive solutions of abstract fized point equations: mushrooms, loops and isolas, J. Diff. Eqns. 209 (2005), 416–441.

    Article  Google Scholar 

  30. 30.

    J. López-Gómez and M. Molina-Meyer, Superlinear indefinite systems: Beyond Lotka Volterra models, J. Differ. Eqns. 221 (2006), 343–411.

    MathSciNet  Article  Google Scholar 

  31. 31.

    J. López-Gómez and M. Molina-Meyer, The competitive exclusion principle versus biodiversity through segregation and further adaptation to spatial heterogeneities, Theor. Popul. Biol. 69 (2006), 94–109.

    Article  Google Scholar 

  32. 32.

    J. López-Gómez and M. Molina-Meyer, Modeling coopetition, Math. Comput. Simul. 76 (2007), 132–140.

    MathSciNet  Article  Google Scholar 

  33. 33.

    J. López-Gómez, M. Molina-Meyer and A. Tellini, Intricate dynamics caused by facilitation in competitive environments within polluted habitat patches, Eur. J. Appl. Maths.https://doi.org/10.1017/S0956792513000429(2014), 1–17.

  34. 34.

    J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one node solutions in a class of degenerate bundary value problems, Disc. Cont. Dyn. Sys. B 22 (2017), 923–946.

    MATH  Google Scholar 

  35. 35.

    J. López-Gómez, M. Molina-Meyer and M. Villareal, Numerical coexistence of coexistence states, SIAM J. Numer. Anal. 29 (1992), 1074–1092.

    MathSciNet  Article  Google Scholar 

  36. 36.

    J. López-Gómez, A. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems, Commun. Pure Appl. Anal. 13 (2014), 1–73.

    MathSciNet  Article  Google Scholar 

  37. 37.

    J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate boundary value problems, Adv. Nonl. Studies 15 (2015), 253–288.

    MathSciNet  MATH  Google Scholar 

  38. 38.

    J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVPs, Top. Meth. Nonl. Anal. 49 (2017), 359–376.

    MathSciNet  MATH  Google Scholar 

  39. 39.

    J. López-Gómez and P. H. Rabinowitz, The structure of the set of 1-node solutions of a class of degenerate BVP’s, J. Diff. Eqns. 268 (2020), 4691–4732.

    MathSciNet  Article  Google Scholar 

  40. 40.

    J. Mawhin, D. Papini and F. Zanolin, Boundary blow-up for differential equations with indefinite weight, J. Diff. Eqns. 188 (2003), 33–51.

    MathSciNet  Article  Google Scholar 

  41. 41.

    P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.

    MathSciNet  Article  Google Scholar 

  42. 42.

    P. H. Rabinowitz, A note on a nonlinear eigenvalue problem for a class of differential equations, J. Diff. Eqns. 9 (1971), 536–548.

    MathSciNet  Article  Google Scholar 

  43. 43.

    P. H. Rabinowitz, A note on a pair of solutions of a nonlinear Sturm–Liouville problem, Manuscr. Math. 11 (1974), 273–282.

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Acknowledgements

We thank the, anonymous, reviewer for his/her extremely careful reading of the paper, which has greatly improved it.

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Correspondence to J. López-Gómez.

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This paper is dedicated to M. Hieber at the occasion of his 60th birthday mit Wertschätzung und Freundschaft.

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Partially supported by the Research Grant PGC2018-097104-B-I00 of the Spanish Ministry of Science, Innovation and Universities, and the Institute of Inter-disciplinary Mathematics (IMI) of Complutense University. M. Fencl has been supported by the Project SGS-2019-010 of the University of West Bohemia, the Project 18-03253S of the Grant Agency of the Czech Republic and the Project LO1506 of the Czech Ministry of Education, Youth and Sport.

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Fencl, M., López-Gómez, J. Nodal solutions of weighted indefinite problems. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00625-7

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Keywords

  • Superlinear indefinite problems
  • Weighted problems
  • Positive solutions
  • Nodal solutions
  • Eigencurves
  • Concavity
  • Bifurcation
  • Global components
  • Path-following
  • Pseudo-spectral methods
  • Finite-difference scheme

Mathematics Subject Classification

  • 34B15
  • 34B08
  • 34L16