On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces

Patrizia Pucci; Letizia Temperini; Patrizia Pucci; Letizia Temperini

doi:10.3934/mine.2023007

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-21. doi: 10.3934/mine.2023007

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On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces

Patrizia Pucci ^{1
,
,},
Letizia Temperini ²

1.
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
2.
Dipartimento di Matematica e Informatica 'Ulisse Dini', Università degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy

Received: 09 September 2021 Revised: 28 December 2021 Accepted: 28 December 2021 Published: 20 January 2022

In this paper we establish some variants of the celebrated concentration–compactness principle of Lions – CC principle briefly – in the classical and fractional Folland–Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland–Stein space, involving the Hardy–Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland–Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.
- Heisenberg group,
- concentration–compactness principles,
- critical exponents,
- Hardy terms,
- integro–differential operators
Citation: Patrizia Pucci, Letizia Temperini. On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces[J]. Mathematics in Engineering, 2023, 5(1): 1-21. doi: 10.3934/mine.2023007

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Abstract

In this paper we establish some variants of the celebrated concentration–compactness principle of Lions – CC principle briefly – in the classical and fractional Folland–Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland–Stein space, involving the Hardy–Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland–Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.

References

[1]	Adimurth, A. Mallick, A Hardy type inequality on fractional order Sobolev spaces on the Heisenberg group, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), ⅩⅧ (2018), 917–949. http://dx.doi.org/10.2422/2036-2145.201604_010 doi: 10.2422/2036-2145.201604_010
[2]	T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom., 11 (1976), 573–598. http://dx.doi.org/10.4310/jdg/1214433725 doi: 10.4310/jdg/1214433725
[3]	G. Bianchi, J. Chabrowski, A. Szulkin, On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlinear Anal. Theor., 25 (1995), 41–59. http://dx.doi.org/10.1016/0362-546X(94)E0070-W doi: 10.1016/0362-546X(94)E0070-W
[4]	J. F. Bonder, N. Saintier, A. Silva, The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis-Nirenberg problem, Nonlinear Differ. Equ. Appl., 25 (2018), 52. http://dx.doi.org/10.1007/s00030-018-0543-5 doi: 10.1007/s00030-018-0543-5
[5]	S. Bordoni, R. Filippucci, P. Pucci, Existence of solutions in problems on Heisenberg groups involving Hardy and critical terms, J. Geom. Anal., 30 (2020), 1887–1917. http://dx.doi.org/10.1007/s12220-019-00295-z doi: 10.1007/s12220-019-00295-z
[6]	J. Chabrowski, Concentration–compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var., 3 (1995), 493–512. http://dx.doi.org/10.1007/BF01187898 doi: 10.1007/BF01187898
[7]	L. D'Ambrosio, Hardy–type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Ⅳ (2005), 451–486. http://dx.doi.org/10.2422/2036-2145.2005.3.04 doi: 10.2422/2036-2145.2005.3.04
[8]	D. Danielli, N. Garofalo, N. C. Phuc, Inequalities of Hardy–Sobolev type in Carnot–Carathéodory, In: Sobolev spaces in mathematics I, New York, NY: Springer, 2009,117–151. http://dx.doi.org/10.1007/978-0-387-85648-3_5
[9]	E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. http://dx.doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
[10]	S. Dipierro, M. Medina, E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb R^n$, Pisa: Edizioni della Normale, 2017.
[11]	N. Dungey, A. F. M. ter Elst, D. W. Robinson, Analysis on Lie groups with polynomial growth, Boston: Birkhäuser Boston, Inc., 2003. http://dx.doi.org/10.1007/978-1-4612-2062-6
[12]	A. Fiscella, P. Pucci, Kirchhoff Hardy fractional problems with lack of compactness, Adv. Nonlinear Stud. 17 (2017), 429–456. http://dx.doi.org/10.1515/ans-2017-6021 doi: 10.1515/ans-2017-6021
[13]	G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161–207. http://dx.doi.org/10.1007/BF02386204 doi: 10.1007/BF02386204
[14]	G. B. Folland, E. M. Stein, Estimates for the $\overline \partial_b$ complex and analysis on the Heisenberg group, Commun. Pure Appl. Math., 27 (1974), 429–522. http://dx.doi.org/10.1002/cpa.3160270403 doi: 10.1002/cpa.3160270403
[15]	I. Fonseca, G. Leoni, Modern methods in the calculus of variations: $L^p$ spaces, New York: Springer, 2007. http://dx.doi.org/10.1007/978-0-387-69006-3
[16]	N. Garofalo, E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier, 40 (1990), 313–356. http://dx.doi.org/10.5802/aif.1215 doi: 10.5802/aif.1215
[17]	N. Garofalo, D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces, Commun. Pure Appl. Math., 49 (1996), 1081–1144.
[18]	J. A. Goldstein, Q. S. Zhang, On a degenerate heat equation with a singular potential, J. Funct. Anal., 186 (2001), 342–359. http://dx.doi.org/10.1006/jfan.2001.3792 doi: 10.1006/jfan.2001.3792
[19]	L. Hőrmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147–171. http://dx.doi.org/10.1007/BF02392081 doi: 10.1007/BF02392081
[20]	S. P. Ivanov, D. N. Vassilev, Extremals for the Sobolev inequality and the quaternionic contact Yamabe problem, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2011. http://dx.doi.org/10.1142/7647
[21]	D. Jerison, J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, In: Microlocal analysis, Providence, RI: Amer. Math. Soc., 1984, 57–63.
[22]	D. Jerison, J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom., 25 (1987), 167–197. http://dx.doi.org/10.4310/jdg/1214440849 doi: 10.4310/jdg/1214440849
[23]	D. Jerison, J. M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc., 1 (1988), 1–13. http://dx.doi.org/10.1090/S0894-0347-1988-0924699-9 doi: 10.1090/S0894-0347-1988-0924699-9
[24]	D. Jerison, J. M. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom., 29 (1989), 303–343. http://dx.doi.org/10.4310/jdg/1214442877 doi: 10.4310/jdg/1214442877
[25]	P. L. Lions, The concentration–compactness principle in the calculus of variations. The limit case, Part 1, Rev. Mat. Iberoamericana, 1 (1985), 145–201. http://dx.doi.org/ 10.4171/RMI/6 doi: 10.4171/RMI/6
[26]	P. L. Lions, The concentration–compactness principle in the calculus of variations. The limit case, Part 2, Rev. Mat. Iberoamericana, 1 (1985), 45–121. http://dx.doi.org/10.4171/RMI/12 doi: 10.4171/RMI/12
[27]	G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana, 8 (1992), 367–439. http://dx.doi.org/10.4171/RMI/129 doi: 10.4171/RMI/129
[28]	S. Mosconi, K. Perera, M. Squassina, Y. Yang, The Brezis–Nirenberg problem for the fractional p-Laplacian, Calc. Var., 55 (2016), 105. http://dx.doi.org/10.1007/s00526-016-1035-2 doi: 10.1007/s00526-016-1035-2
[29]	P. Niu, H. Zhang, Y. Wang, Hardy–type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc., 129 (2001), 3623–3630. http://dx.doi.org/10.2307/2699514 doi: 10.2307/2699514
[30]	G. Palatucci, A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var., 50 (2014), 799–829. http://dx.doi.org/10.1007/s00526-013-0656-y doi: 10.1007/s00526-013-0656-y
[31]	P. Pucci, L. Temperini, Existence for $(p, q)$ critical systems in the Heisenberg group, Adv. Nonlinear Anal., 9 (2020), 895–922. http://dx.doi.org/10.1515/anona-2020-0032 doi: 10.1515/anona-2020-0032
[32]	P. Pucci, L. Temperini, Concentration–compactness results for systems in the Heisenberg group, Opuscula Math., 40 (2020), 151–162. http://dx.doi.org/10.7494/OpMath.2020.40.1.151 doi: 10.7494/OpMath.2020.40.1.151
[33]	P. Pucci, L. Temperini, Existence for fractional $(p, q)$ systems with critical and Hardy terms in $\mathbb R^N$, Nonlinear Anal., 211 (2021), 112477. http://dx.doi.org/10.1016/j.na.2021.112477 doi: 10.1016/j.na.2021.112477
[34]	G. Talenti, Best constant in Sobolev inequality, Annali di Matematica, 110 (1976), 353–372. http://dx.doi.org/10.1007/BF02418013 doi: 10.1007/BF02418013
[35]	D. Vassilev, Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups, Pac. J. Math., 227 (2006), 361–397. http://dx.doi.org/10.2140/pjm.2006.227.361 doi: 10.2140/pjm.2006.227.361

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