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On Symmetric Solutions for (p, q)-Laplacian Equations in \({\mathbb {R}}^N\) with Critical Terms

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Abstract

We prove existence and multiplicity results in \(\mathbb R^N\) for an elliptic problem of (p, q)-Laplacian type with a nonlinearity involving both a critical term and a subcritical term with a positive real parameter \(\lambda \). In particular, nonnegative nontrivial weights satisfying some symmetry conditions with respect to a certain group T are included in the nonlinearity. We prove first the existence of at least one solution with positive energy for \(\lambda \) sufficiently small using Mountain Pass Theorem, then we obtain the existence of infinitely many weak solutions with positive (finite) energy for every \(\lambda \) positive applying Fountain Theorem. Our proofs use variational methods and concentration compactness principles.

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Acknowledgements

LB and RF 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). RF was partly supported by Fondo Ricerca di Base di Ateneo Esercizio 2017-19 of the University of Perugia, named Problemi con non linearità dipendenti dal gradiente and by INdAM-GNAMPA Project 2020 titled Equazioni alle derivate parziali: problemi e modelli (Prot_U-UFMBAZ-2020).

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  1. Department of Mathematics, University of Perugia, Via Vanvitelli 1, 06123, Perugia, Italy

    Ylenia Brizi & Roberta Filippucci

  2. Department of Mathematics, University of Firenze, Viale Morgagni 40-44, 50134, Firenze, Italy

    Laura Baldelli

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  1. Laura Baldelli
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Baldelli, L., Brizi, Y. & Filippucci, R. On Symmetric Solutions for (p, q)-Laplacian Equations in \({\mathbb {R}}^N\) with Critical Terms. J Geom Anal 32, 120 (2022). https://doi.org/10.1007/s12220-021-00846-3

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