Home Existence for singular critical exponential (𝑝, 𝑄) equations in the Heisenberg group
Article
Licensed
Unlicensed Requires Authentication

Existence for singular critical exponential (𝑝, 𝑄) equations in the Heisenberg group

  • Patrizia Pucci ORCID logo EMAIL logo and Letizia Temperini ORCID logo
Published/Copyright: May 26, 2021

Abstract

The paper deals with the existence of nontrivial solutions for ( p , Q ) equations in the Heisenberg group H n with critical exponential growth at infinity and a singular behavior at the origin. The main features and novelty of the paper are the above generality on the right-hand side of the equation, the ( p , Q ) growth of the elliptic operator and the fact that the equation is studied in the entire Heisenberg group.

MSC 2010: 35B08; 35B33; 35J20; 35J60; 35R03

Award Identifier / Grant number: Prot_U-UFMBAZ-2020-000761

Funding statement: P. Pucci and L. Temperini 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). The manuscript was realized within the auspices of the INdAM-GNAMPA Project 2020 titled Equazioni alle derivate parziali: problemi e modelli (Prot_U-UFMBAZ-2020-000761). P. Pucci was also partly supported by the Fondo Ricerca di Base di Ateneo – Esercizio 2017–2019 of the University of Perugia, named PDEs and Nonlinear Analysis.

  1. Communicated by: Juan Manfredi

References

[1] S. Adachi and K. Tanaka, Trudinger type inequalities in R N and their best exponents, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2051–2057. 10.1090/S0002-9939-99-05180-1Search in Google Scholar

[2] D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2) 128 (1988), no. 2, 385–398. 10.2307/1971445Search in Google Scholar

[3] C. O. Alves and L. R. de Freitas, Multiplicity of nonradial solutions for a class of quasilinear equations on annulus with exponential critical growth, Topol. Methods Nonlinear Anal. 39 (2012), no. 2, 243–262. Search in Google Scholar

[4] C. O. Alves and L. R. de Freitas, Multiplicity results for a class of quasilinear equations with exponential critical growth, Math. Nachr. 291 (2018), no. 2–3, 222–224. 10.1002/mana.201500371Search in Google Scholar

[5] C. O. Alves, L. R. de Freitas and S. H. M. Soares, Indefinite quasilinear elliptic equations in exterior domains with exponential critical growth, Differential Integral Equations 24 (2011), no. 11–12, 1047–1062. 10.57262/die/1356012875Search in Google Scholar

[6] A. Baldi, B. Franchi, N. Tchou and M. C. Tesi, Compensated compactness for differential forms in Carnot groups and applications, Adv. Math. 223 (2010), no. 5, 1555–1607. 10.1016/j.aim.2009.09.020Search in Google Scholar

[7] Z. M. Balogh and A. Kristály, Lions-type compactness and Rubik actions on the Heisenberg group, Calc. Var. Partial Differential Equations 48 (2013), no. 1–2, 89–109. 10.1007/s00526-012-0543-ySearch in Google Scholar

[8] S. Bordoni and P. Pucci, Schrödinger–Hardy systems involving two Laplacian operators in the Heisenberg group, Bull. Sci. Math. 146 (2018), 50–88. 10.1016/j.bulsci.2018.03.001Search in Google Scholar

[9] D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R 2 , Comm. Partial Differential Equations 17 (1992), no. 3–4, 407–435. 10.1080/03605309208820848Search in Google Scholar

[10] L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progr. Math. 259, Birkhäuser, Basel, 2007. Search in Google Scholar

[11] P. Cartier, Quantum mechanical commutation relations and theta functions, Algebraic Groups and Discontinuous Subgroups, American Mathematical Society, Providence (1966), 361–383. 10.1090/pspum/009/0216825Search in Google Scholar

[12] L. Cherfils and Y. Ilýasov, On the stationary solutions of generalized reaction diffusion equations with p & q –Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9–22. 10.3934/cpaa.2005.4.9Search in Google Scholar

[13] W. S. Cohn, N. Lam, G. Lu and Y. Yang, The Moser–Trudinger inequality in unbounded domains of Heisenberg group and sub-elliptic equations, Nonlinear Anal. 75 (2012), no. 12, 4483–4495. 10.1016/j.na.2011.09.053Search in Google Scholar

[14] W. S. Cohn and G. Lu, Best constants for Moser-Trudinger inequalities on the Heisenberg group, Indiana Univ. Math. J. 50 (2001), no. 4, 1567–1591. 10.1512/iumj.2001.50.2138Search in Google Scholar

[15] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), no. 1, 219–273. 10.1007/s00205-015-0859-9Search in Google Scholar

[16] M. Colombo and G. Mingione, Calderón–Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal. 270 (2016), no. 4, 1416–1478. 10.1016/j.jfa.2015.06.022Search in Google Scholar

[17] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in R 2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 139–153. 10.1007/BF01205003Search in Google Scholar

[18] P. P. Divakaran, Quantum theory as the representation theory of symmetries, Phys. Rev. Lett. 79 (1997), no. 12, 2159–2163. 10.1103/PhysRevLett.79.2159Search in Google Scholar

[19] J. A. M. B. do Ó, 𝑁-Laplacian equations in R N with critical growth, Abstr. Appl. Anal. 2 (1997), no. 3–4, 301–315. 10.1155/S1085337597000419Search in Google Scholar

[20] G. M. Figueiredo and F. B. M. Nunes, Existence of positive solutions for a class of quasilinear elliptic problems with exponential growth via the Nehari manifold method, Rev. Mat. Complut. 32 (2019), no. 1, 1–18. 10.1007/s13163-018-0283-4Search in Google Scholar

[21] A. Fiscella and P. Pucci, ( p , q ) systems with critical terms in R N , Nonlinear Anal. 177 (2018), 454–479. 10.1016/j.na.2018.03.012Search in Google Scholar

[22] A. Fiscella and P. Pucci, ( p , N ) equations with critical exponential nonlinearities in R N , J. Math. Anal. Appl. 501 (2021), Article ID 123379. Search in Google Scholar

[23] A. Fiscella and P. Pucci, Degenerate Kirchhoff ( p , q ) -fractional systems with critical nonlinearities, Fract. Calc. Appl. Anal. 23 (2020), no. 3, 723–752. 10.1515/fca-2020-0036Search in Google Scholar

[24] G. B. Folland and E. M. Stein, Estimates for the ¯ b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. 10.1002/cpa.3160270403Search in Google Scholar

[25] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton University, Princeton, 1982. 10.1515/9780691222455Search in Google Scholar

[26] B. Franchi, G. Lu and R. L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 577–604. 10.5802/aif.1466Search in Google Scholar

[27] N. Fusco and C. Sbordone, Local boundedness of minimizers in a limit case, Manuscripta Math. 69 (1990), no. 1, 19–25. 10.1007/BF02567909Search in Google Scholar

[28] N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 313–356. 10.5802/aif.1215Search in Google Scholar

[29] N. Garofalo and D.-M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), no. 10, 1081–1144. 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-ASearch in Google Scholar

[30] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. 10.1007/BF02392081Search in Google Scholar

[31] S. P. Ivanov and D. N. Vassilev, Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem, World Scientific, Hackensack, 2011. 10.1142/7647Search in Google Scholar

[32] E. Kaniuth, A Course in Commutative Banach Algebras, Grad. Texts in Math. 246, Springer, New York, 2009. 10.1007/978-0-387-72476-8Search in Google Scholar

[33] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of 𝑁-Laplacian type with critical exponential growth in R N , J. Funct. Anal. 262 (2012), no. 3, 1132–1165. 10.1016/j.jfa.2011.10.012Search in Google Scholar

[34] 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

[35] N. Lam and G. Lu, 𝑁-Laplacian equations in R N with subcritical and critical growth without the Ambrosetti–Rabinowitz condition, Adv. Nonlinear Stud. 13 (2013), no. 2, 289–308. 10.1515/ans-2013-0203Search in Google Scholar

[36] N. Lam, G. Lu and H. Tang, On nonuniformly subelliptic equations of 𝑄-sub-Laplacian type with critical growth in the Heisenberg group, Adv. Nonlinear Stud. 12 (2012), no. 3, 659–681. 10.1515/ans-2012-0312Search in Google Scholar

[37] N. Lam, G. Lu and H. Tang, Sharp subcritical Moser–Trudinger inequalities on Heisenberg groups and subelliptic PDEs, Nonlinear Anal. 95 (2014), 77–92. 10.1016/j.na.2013.08.031Search in Google Scholar

[38] G. P. Leonardi and S. Masnou, On the isoperimetric problem in the Heisenberg group H n , Ann. Mat. Pura Appl. (4) 184 (2005), no. 4, 533–553. 10.1007/s10231-004-0127-3Search in Google Scholar

[39] J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 (2018), no. 3, Paper No. 84. 10.1007/s00526-018-1352-8Search in Google Scholar

[40] G. Lu, Existence and size estimates for the Green’s functions of differential operators constructed from degenerate vector fields, Comm. Partial Differential Equations 17 (1992), no. 7–8, 1213–1251. 10.1080/03605309208820883Search in Google Scholar

[41] G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications, Rev. Mat. Iberoam. 8 (1992), no. 3, 367–439. 10.4171/RMI/129Search in Google Scholar

[42] G. Lu, The sharp Poincaré inequality for free vector fields: An endpoint result, Rev. Mat. Iberoam. 10 (1994), no. 2, 453–466. 10.4171/RMI/158Search in Google Scholar

[43] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 5, 391–409. 10.1016/s0294-1449(16)30379-1Search in Google Scholar

[44] G. Mingione, A. Zatorska-Goldstein and X. Zhong, Gradient regularity for elliptic equations in the Heisenberg group, Adv. Math. 222 (2009), no. 1, 62–129. 10.1016/j.aim.2009.03.016Search in Google Scholar

[45] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. 10.1512/iumj.1971.20.20101Search in Google Scholar

[46] P. Pucci and L. Temperini, Existence for ( p , q ) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), no. 1, 895–922. 10.1515/anona-2020-0032Search in Google Scholar

[47] P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional 𝑝-Laplacian in R N , Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2785–2806. 10.1007/s00526-015-0883-5Search in Google Scholar

[48] J. Simon, Régularité de la solution d’une équation non linéaire dans R N , Journées d’Analyse Non Linéaire, Lecture Notes in Math. 665, Springer, Berlin (1978), 205–227. 10.1007/BFb0061807Search in Google Scholar

[49] K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College, London, 2007. 10.1142/p456Search in Google Scholar

[50] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. 10.1512/iumj.1968.17.17028Search in Google Scholar

[51] Y. Yang, Trudinger–Moser inequalities on the entire Heisenberg group, Math. Nachr. 287 (2014), no. 8–9, 1071–1080. 10.1002/mana.201200019Search in Google Scholar

[52] Y. Yang and K. Perera, ( N , q ) -Laplacian problems with critical Trudinger–Moser nonlinearities, Bull. Lond. Math. Soc. 48 (2016), no. 2, 260–270. 10.1112/blms/bdw002Search in Google Scholar

[53] S. Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 1, 305–363. 10.5802/aif.1568Search in Google Scholar

[54] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710. 10.1070/IM1987v029n01ABEH000958Search in Google Scholar

Received: 2020-03-28
Revised: 2021-03-31
Accepted: 2021-04-20
Published Online: 2021-05-26
Published in Print: 2022-07-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 23.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/acv-2020-0028/html
Scroll to top button