Abstract
We investigate some of the effects of the lack of compactness in the critical Folland–Stein–Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi’s Γ-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe [M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 1984, 4, 511–517].
Funding statement: The authors are supported by INdAM projects “Fenomeni non locali in problemi locali”, Grant No. CUP_E55F22000270001. The first two authors are supported by “Problemi non locali: teoria cinetica e non uniforme ellitticità”, Grant No. CUP_E53C22001930001 and “Problemi ellittici e sub-ellittici: singolarità e crescita critica”, Grant No. CUP_E53C23001670001. The second author is also supported by the Project “Local vs Nonlocal: mixed-type operators and nonuniform ellipticity”, Grant No. CUP_D91B21005370003. The results in this paper have been announced in the preliminary research report .
References
[1] M. Amar and A. Garroni, Γ-convergence of concentration problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 1, 151–179. Search in Google Scholar
[2] J. Benameur, Description du défaut de compacité de l’injection de Sobolev sur le groupe de Heisenberg, Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 4, 599–624. 10.36045/bbms/1225893942Search in Google Scholar
[3] T. P. Branson, L. Fontana and C. Morpurgo, Moser–Trudinger and Beckner–Onofri’s inequalities on the CR sphere, Ann. of Math. (2) 177 (2013), no. 1, 1–52. 10.4007/annals.2013.177.1.1Search in Google Scholar
[4] H. Brezis and L. A. Peletier, Asymptotics for elliptic equations involving critical growth, Partial Differential Equations and the Calculus of Variations, Vol. I, Progr. Nonlinear Differential Equations Appl. 1, Birkhäuser, Boston (1989), 149–192. 10.1007/978-1-4684-9196-8_7Search in Google Scholar
[5] G. Catino, Y. Li, D. Monticelli and A. Roncoroni, A Liouville theorem in the Heisenberg group, preprint (2023), https://arxiv.org/abs/2310.10469. Search in Google Scholar
[6] G. Citti, Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian, Ann. Mat. Pura Appl. (4) 169 (1995), 375–392. 10.1007/BF01759361Search in Google Scholar
[7] G. Citti and F. Uguzzoni, Critical semilinear equations on the Heisenberg group: The effect of the topology of the domain, Nonlinear Anal. 46 (2001), no. 3, 399–417. 10.1016/S0362-546X(00)00138-3Search in Google Scholar
[8]
D. Danielli, N. Garofalo and A. Petrosyan,
The sub-elliptic obstacle problem:
[9] M. del Pino, M. Musso and F. Pacard, Bubbling along boundary geodesics near the second critical exponent, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 6, 1553–1605. 10.4171/jems/241Search in Google Scholar
[10] S. Deng, F. Mahmoudi and M. Musso, Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents, Proc. Lond. Math. Soc. (3) 118 (2019), no. 2, 379–415. 10.1112/plms.12183Search in Google Scholar
[11] J. Flynn and J. Vétois, Liouville-type results for the CR Yamabe equation in the Heisenberg group, preprint (2023), https://arxiv.org/abs/2310.14048. 10.2422/2036-2145.202311_016Search in Google Scholar
[12] M. Flucher and S. Müller, Concentration of low energy extremals, Ann. Inst. H. Poincaré C Anal. Non Linéaire 16 (1999), no. 3, 269–298. 10.1016/s0294-1449(99)80015-8Search in Google Scholar
[13]
G. B. Folland and E. M. Stein,
Estimates for the
[14] R. L. Frank, M. d. M. González, D. D. Monticelli and J. Tan, An extension problem for the CR fractional Laplacian, Adv. Math. 270 (2015), 97–137. 10.1016/j.aim.2014.09.026Search in Google Scholar
[15] R. L. Frank and E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. of Math. (2) 176 (2012), no. 1, 349–381. 10.4007/annals.2012.176.1.6Search in Google Scholar
[16] N. Garofalo and E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41 (1992), no. 1, 71–98. 10.1512/iumj.1992.41.41005Search in Google Scholar
[17] N. Garofalo and D. Vassilev, Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups, Math. Ann. 318 (2000), no. 3, 453–516. 10.1007/s002080000127Search in Google Scholar
[18] N. Garofalo and D. Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type, Duke Math. J. 106 (2001), no. 3, 411–448. 10.1215/S0012-7094-01-10631-5Search in Google Scholar
[19] P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233. 10.1051/cocv:1998107Search in Google Scholar
[20] C. Guidi, A. Maalaoui and V. Martino, Palais–Smale sequences for the fractional CR Yamabe functional and multiplicity results, Calc. Var. Partial Differential Equations 57 (2018), no. 6, Paper No. 152. 10.1007/s00526-018-1423-xSearch in Google Scholar
[21] Z.-C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré C Anal. Non Linéaire 8 (1991), no. 2, 159–174. 10.1016/s0294-1449(16)30270-0Search in Google Scholar
[22] S. P. Ivanov and D. N. Vassilev, Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem, World Scientific, Hackensack, 2011. 10.1142/9789814295710Search in Google Scholar
[23] D. Jerison and J. M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), no. 1, 1–13. 10.1090/S0894-0347-1988-0924699-9Search in Google Scholar
[24] N. Lam and G. Lu, Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math. 231 (2012), no. 6, 3259–3287. 10.1016/j.aim.2012.09.004Search in Google Scholar
[25] E. Lanconelli and F. Uguzzoni, Asymptotic behavior and non-existence theorems for semilinear Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 1, 139–168. Search in Google Scholar
[26] A. Maalaoui, V. Martino and A. Pistoia, Concentrating solutions for a sub-critical sub-elliptic problem, Differential Integral Equations 26 (2013), no. 11–12, 1263–1274. 10.57262/die/1378327425Search in Google Scholar
[27] M. Manfredini, G. Palatucci, M. Piccinini and S. Polidoro, Hölder continuity and boundedness estimates for nonlinear fractional equations in the Heisenberg group, J. Geom. Anal. 33 (2023), no. 3, Paper No. 77. 10.1007/s12220-022-01124-6Search in Google Scholar
[28] G. Palatucci, p-Laplacian problems with critical Sobolev exponent, Asymptot. Anal. 73 (2011), no. 1–2, 37–52. 10.3233/ASY-2010-1029Search in Google Scholar
[29] G. Palatucci, Subcritical approximation of the Sobolev quotient and a related concentration result, Rend. Semin. Mat. Univ. Padova 125 (2011), 1–14. 10.4171/rsmup/125-1Search in Google Scholar
[30] G. Palatucci and M. Piccinini, Nonlocal Harnack inequalities in the Heisenberg group, Calc. Var. Partial Differential Equations 61 (2022), no. 5, Paper No. 185. 10.1007/s00526-022-02301-9Search in Google Scholar
[31] G. Palatucci and M. Piccinini, Asymptotic approach to singular solutions for the CR Yamabe equation, preprint (2024), https://arxiv.org/abs/2307.14933v2. Search in Google Scholar
[32] G. Palatucci and M. Piccinini, Nonlinear fractional equations in the Heisenberg group, Bruno Pini Mathematical Analysis Seminar 2023, University of Bologna, Bologna (2024), 163–200. Search in Google Scholar
[33] G. Palatucci, M. Piccinini and L. Temperini, Global compactness, subcritical approximation of the Sobolev quotient, and a related concentration result in the Heisenberg group, Extended Abstracts 2021/2022—Methusalem Lectures, Trends Math., Birkhäuser/Springer, Cham (2024), 145–155. 10.1007/978-3-031-48579-4_15Search in Google Scholar
[34] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations 50 (2014), no. 3–4, 799–829. 10.1007/s00526-013-0656-ySearch in Google Scholar
[35] G. Palatucci and A. Pisante, A global compactness type result for Palais–Smale sequences in fractional Sobolev spaces, Nonlinear Anal. 117 (2015), 1–7. 10.1016/j.na.2014.12.027Search in Google Scholar
[36] G. Palatucci, A. Pisante and Y. Sire, Subcritical approximation of a Yamabe-type nonlocal equation: A gamma-convergence approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 3, 819–840. 10.2422/2036-2145.201302_006Search in Google Scholar
[37] D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal. 114 (1993), no. 1, 97–105. 10.1006/jfan.1993.1064Search in Google Scholar
[38]
J. V. Prajapat and A. S. Varghese,
Symmetry and classification of solutions to an integral equation in the Heisenberg group
[39] P. Pucci and L. Temperini, On the concentration-compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), no. 1, 1–21. 10.3934/mine.2023007Search in Google Scholar
[40] O. Rey, Proof of two conjectures of H. Brézis and L. A. Peletier, Manuscripta Math. 65 (1989), no. 1, 19–37. 10.1007/BF01168364Search in Google Scholar
[41] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), no. 4, 511–517. 10.1007/BF01174186Search in Google Scholar
[42] C. Tintarev, Concentration Compactness—Functional-Analytic Theory of Concentration Phenomena, De Gruyter Ser. Nonlinear Anal. Appl. 33, De Gruyter, Berlin, 2020. 10.1515/9783110532432Search in Google Scholar
[43] K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College, London, 2007. 10.1142/9781860947971Search in Google Scholar
[44] F. Uguzzoni, A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group, NoDEA Nonlinear Differential Equations Appl. 6 (1999), no. 2, 191–206. 10.1007/s000300050072Search in Google Scholar
[45] D. Vassilev, Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups, Pacific J. Math. 227 (2006), no. 2, 361–397. 10.2140/pjm.2006.227.361Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
Articles in the same Issue
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case