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Leray–Lions Equations of (p, q)-Type in the Entire Space with Unbounded Potentials

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Abstract

In this paper we prove the existence of signed bounded solutions for a quasilinear elliptic equation in \({\mathbb {R}}^N\) driven by a Leray–Lions operator of (p, q)–type in presence of unbounded potentials. A direct approach seems to be a hard task, and for this reason we will study approximating problems in bounded domains, whose resolutions needs refined tools from nonlinear analysis. In particular, we will use a weaker version of the classical Cerami–Palais–Smale condition together with a extension of the Weierstrass Theorem due to Candela–Palmieri, as well as a generalization of a celebrated convergence result by Boccardo–Murat–Puel.

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Acknowledgements

The authors wish to thank the anonymous referee for her/his extremely careful report, which has highly improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125, Bari, Italy

    Federica Mennuni

  2. Department of Ecological and Biological Sciences, University of Tuscia, Largo dell’Università, 01100, Viterbo, Italy

    Dimitri Mugnai

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  1. Federica Mennuni
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  2. Dimitri Mugnai
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Correspondence to Dimitri Mugnai.

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F.M. is a member of the Research Group INdAM-GNAMPA. D.M. is a member of the Research Group INdAM-GNAMPA. He is supported by the GNAMPA Project 2023 Variational and non-variational problems with lack of compactness CUP_E53C22001930001, and by the Fondo di finanziamento per le attività base di ricerca (FFABR) 2017.

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Mennuni, F., Mugnai, D. Leray–Lions Equations of (p, q)-Type in the Entire Space with Unbounded Potentials. Milan J. Math. (2024). https://doi.org/10.1007/s00032-024-00391-y

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