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Existence Problems on Heisenberg Groups Involving Hardy and Critical Terms

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Abstract

In this paper, we study existence and asymptotic behavior of nontrivial solutions of a series of problems in general open subsets \(\Omega \) of the Heisenberg group \(\mathbb {H}^n\), possibly unbounded or even \(\mathbb {H}^n\). The problems involve the p-Laplacian operator on \(\mathbb {H}^n\), a Hardy coefficient and different critical nonlinearities. The main originality of the paper is to work in the Heisenberg group.

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Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM—GNAMPA Projects Problemi non lineari alle derivate parziali (Prot_U-UFMBAZ-2018-000384). R. Filippucci and P. Pucci were partly supported by the Italian MIUR project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT_009) and by the Fondo Ricerca di Base di Ateneo—Esercizio 2015 of the University of Perugia, named Non esistenza di soluzioni intere and PDEs e Analisi Nonlineare, respectively.

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  1. Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/a, 50134, Florence, Italy

    Sara Bordoni

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123, Perugia, Italy

    Sara Bordoni, Roberta Filippucci & Patrizia Pucci

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  1. Sara Bordoni
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  2. Roberta Filippucci
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Correspondence to Patrizia Pucci.

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Bordoni, S., Filippucci, R. & Pucci, P. Existence Problems on Heisenberg Groups Involving Hardy and Critical Terms. J Geom Anal 30, 1887–1917 (2020). https://doi.org/10.1007/s12220-019-00295-z

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