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Existence of positive eigenfunctions to an anisotropic elliptic operator via the sub-supersolution method

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Abstract

Using the sub-supersolution method, we study the existence of positive solutions for the anisotropic problem

$$\begin{aligned} -\sum _{i=1}^N\frac{\partial }{\partial x_i}\left( \left| \frac{\partial u}{\partial x_i}\right| ^{p_i-2}\frac{\partial u}{\partial x_i}\right) =\lambda u^{q-1} \end{aligned}$$
(0.1)

where \(\Omega \) is a bounded and regular domain of \({\mathbb {R}}^N\), \(q>1\), and \(\lambda >0\).

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Acknowledgements

The authors wish to thank professors A. Cianchi and V. Vespri for enlightening conversations about the problem. Moreover, the authors acknowledge IMUS (Mathematics Institute of the University of Seville), IEMath-GR (Mathematics Institute of the University of Granada), and University of Cadiz for supporting a PhD course from which the collaboration arose. S. Ciani is partially founded by INdAM group GNAMPA, A. Suárez by PGC2018-098308-B-I00 (MCI/AEI/FEDER, UE), and G.M. Figueiredo by CNPQ, CAPES, and FAPDF. Finally, we would like to acknowledge the anonymous referee for her/his constructive criticism which have increased the value of the present paper and stimulated curiosity towards new open problems.

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Authors and Affiliations

  1. Dpto. di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, viale G. Morgagni 67/A, 50134, Firenze, Italy

    Simone Ciani

  2. Dpto. de Matemática, Universidade de Brasília, UNB, CEP: 70910-900, Brasília-DF, Brazil

    Giovany M. Figueiredo

  3. Dpto. EDAN and IMUS, University of Sevilla, Avda. Reina Mercedes, s/n, 41012, Sevilla, Spain

    Antonio Suárez

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  1. Simone Ciani
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  2. Giovany M. Figueiredo
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  3. Antonio Suárez
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Correspondence to Antonio Suárez.

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Ciani, S., Figueiredo, G.M. & Suárez, A. Existence of positive eigenfunctions to an anisotropic elliptic operator via the sub-supersolution method. Arch. Math. 116, 85–95 (2021). https://doi.org/10.1007/s00013-020-01518-4

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