Opuscula Math. 40, no. 1 (2020), 151-163
https://doi.org/10.7494/OpMath.2020.40.1.151

Opuscula Mathematica

# Concentration-compactness results for systems in the Heisenberg group

Patrizia Pucci
Letizia Temperini

Abstract. In this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895–922] on some variants of the concentration-compactness principle in bounded PS domains $$\Omega$$ of the Heisenberg group $$\mathbb{H}^n$$. The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms.

Keywords: Heisenberg group, concentration-compactness, critical exponents, Hardy terms.

Mathematics Subject Classification: 22E30, 35B33, 35J50, 58E30, 35H05, 35A23.

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1. S. Bordoni, P. Pucci, Schrödinger-Hardy systems involving two Laplacian operators in the Heisenberg group, Bull. Sci. Math. 146 (2018), 50-88.
2. S. Bordoni, R. Filippucci, P. Pucci, Existence of solutions in problems on Heisenberg groups involving Hardy and critical terms, J. Geom. Anal., Special Issue Perspectives of Geometric Analysis in PDEs (2019) 29 pp., https://doi.org/10.1007/s12220-019-00295-z
3. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
4. L. D'Ambrosio, Hardy-type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 451-486.
5. A. Fiscella, P. Pucci, Kirchhoff Hardy fractional problems with lack of compactness, Adv. Nonlinear Stud. 17 (2017), 429-456.
6. A. Fiscella, P. Pucci, $$(p,q)$$ systems with critical terms in $$\mathbb{R}^N$$\$, Special Issue on Nonlinear PDEs and Geometric Function Theory, in honor of Carlo Sbordone on his 70th birthday, Nonlinear Anal. 177 (2018), Part B, 454-479.
7. G.B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207.
8. G.B. Folland, E.M. Stein, Estimates for the $$\overline \partial_b$$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522.
9. B. Franchi, C. Gutierrez, R.L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), 523-604.
10. N. Garofalo, E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier 40 (1990), 313-356.
11. N. Garofalo, D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081-1144.
12. S.P. Ivanov, D.N. Vassilev, Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
13. D. Kang, Concentration compactness principles for the systems of critical elliptic equations, Differ. Equ. Appl. 4 (2012), 435-444.
14. P.-L. Lions, The concentration compactness principle in the calculus of variations. The limit case. Part 1, Rev. Mat. Iberoamericana 1.1 (1985), 145-201.
15. P.-L. Lions, The concentration compactness principle in the calculus of variations. The limit case. Part 2, Rev. Mat. Iberoamericana 1.2 (1985), 45-121.
16. P. Niu, H. Zhang, Y. Wang, Hardy-type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc. 129 (2001), 3623-3630.
17. P. Pucci, Critical Schrödinger-Hardy systems in the Heisenberg group, Discrete Contin. Dyn. Syst. Ser. S, Special Issue on the occasion of the 60th birthday of Professor Vicentiu D. Radulescu, 12 (2019), 375-400.
18. P. Pucci, Existence of entire solutions for quasilinear equations in the Heisenberg group, Minimax Theory Appl., Special Issue on Nonlinear Phenomena: Theory and Applications 4 (2019), 32 pp.
19. P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895-922.
20. M. Ruzhansky, D. Suragan, Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups, Adv. Math. 317 (2017), 799-822.
21. N. Varopoulos, Analysis on nilpotent Lie groups, J. Funct. Anal. 66 (1986), 406-431.
22. N. Varopoulos, Sobolev inequalities on Lie groups and symmetric spaces, J. Funct. Anal. 86 (1989), 19-40.
23. D. Vassilev, Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups, Pacific J. Math. 227 (2006), 361-397.
24. Q.-H. Yang, Hardy type inequalities related to Carnot-Carathéodory distance on the Heisenberg group, Proc. Amer. Math. Soc. 141 (2013), 351-362.
• Patrizia Pucci (corresponding author)
• https://orcid.org/0000-0001-7242-8485
• Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
• Letizia Temperini
• Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy
• Communicated by P.A. Cojuhari.
• Revised: 2020-01-27.
• Accepted: 2020-01-27.
• Published online: 2020-02-17.