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Nodal solutions of weighted indefinite problems

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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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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 30100, Plzeň, Czech Republic

    M. Fencl

  2. Department of Analysis and Applied Mathematics, Institute of Inter-disciplinary Mathematics (IMI), Complutense University of Madrid, 28040, Madrid, Spain

    J. López-Gómez

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  1. M. Fencl
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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. 21, 2815–2835 (2021). https://doi.org/10.1007/s00028-020-00625-7

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