Nodal solutions of weighted indefinite problems


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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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).

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  • 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