Existence for (p,q) critical systems in the Heisenberg group

Patrizia Pucci,Letizia Temperini

doi:10.1515/anona-2020-0032

Abstract

AbstractThis paper deals with the existence of entire nontrivial solutions for critical quasilinear systems (𝓢) in the Heisenberg group ℍn, driven by general (p,q) elliptic operators of Marcellini types. The study of (𝓢) requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p,q) systems.

Highlights

In recent years, great attention has been focused on the study of (p, q) systems, for their mathematical interest, and for their relevant physical interpretation in applied sciences
This paper deals with the existence of entire nontrivial solutions for critical quasilinear systems (S) in the Heisenberg group Hn, driven by general (p, q) elliptic operators of Marcellini types
It is well known that the Heisenberg group Hn, n =, . . . , appears in various areas, such as quantum theory cf. [1, 2], signal theory cf. [3], theory of theta functions cf. [1, 4], and number theory

Summary

Introduction

Great attention has been focused on the study of (p, q) systems, for their mathematical interest, and for their relevant physical interpretation in applied sciences. It is well known that the Heisenberg group Hn, n = , , . For additional physical interpretations we mention [5], while for general motivations in setting problems in the Heisenberg group context we refer to [6,7,8,9,10,11] and the papers cited there. We prove the existence of nontrivial solutions for quasilinear elliptic systems in the Heisenberg group Hn, involving (p, q) operators, which generalize the ones introduced by Marcellini in [12]. We consider the system in Hn. where λ is a positive real parameter, Q = n + is the homogeneous dimension of the Heisenberg group Hn, α > and β > are two exponents such that α + β = ℘* and ℘* is a critical exponent associated to ℘, with

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The concentration compactness principle for critical systems on Hn

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