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Multiple principal eigenvalues for indefinite quasilinear problems

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Abstract

In this article we study the existence of principal eigenvalues for a completely indefinite problem involving the p-Laplacian:

$$\begin{aligned} -\Delta _ p u + a(x) |u|^{p-2}u = \lambda m(x) |u|^{p-2}u \end{aligned}$$

in an open bounded set with homogeneous Dirichlet boundary conditions. This problem is called “completely indefinite” since both the potential a(x) and the weight m(x) may change sign, so that the “positivity” and coerciveness are lost. The results obtained here generalize an earlier paper written for \(p=2\) (the classical Laplacian) by Fleckinger, Hernández and de Thélin. It extends also some of the results obtained more recently by Cuesta and Ramos-Quoirin. We use two methods: the first one transforms the problem into a “definite” one and defines an eigenvalue depending on a parameter. The second one uses curves of eigenvalues.

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Acknowledgements

The research of J. Hernández is partially supported by the project PID2020-112517GB-I00 Agencia Estatal de Investigación, Spain.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Toulouse 1-Capitole, 31042, Toulouse Cedex, France

    Jacqueline Fleckinger

  2. Instituto de Matemática Interdisciplinar, Facultad de Matemáticas, 28040, Madrid, Spain

    Jesús Hernández

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  1. Jacqueline Fleckinger
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  2. Jesús Hernández
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Correspondence to Jesús Hernández.

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Fleckinger, J., Hernández, J. Multiple principal eigenvalues for indefinite quasilinear problems. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 21 (2023). https://doi.org/10.1007/s13398-022-01349-8

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