Online First version
https://doi.org/10.7494/OpMath.202605211

 
Opuscula Mathematica

Existence of solutions for a doubly critical Schrödinger-Poisson system on the first Heisenberg group

Xueyan Ma
Shaoyun Shi
Yueqiang Song

Abstract. This work is devoted to the study of a class of Schrödinger-Poisson system with doubly critical growth on the first Heisenberg group. Utilizing the concentration-compactness principle associated with classical Sobolev space on the Heisenberg group and mountain pass theorem, we prove that the system admits multiple nontrivial solutions.

Keywords: Heisenberg group, Schrödinger-Poisson system, concentration-compactness principle, mountain pass theorem, nontrivial solutions.

Mathematics Subject Classification: 35A15, 35B33, 47G20.

Full text (pdf)

  1. A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
  2. Y.C. An, H. Liu, The Schrödinger-Poisson type system involving a critical nonlinearity on the first Heisenberg group, Isr. J. Math. 235 (2020), 385-411. https://doi.org/10.1007/s11856-020-1961-8
  3. A. Azzollini, P. D'Avenia, G. Vaira, Generalized Schrödinger-Newton system in dimension \(N\geq3\): critical case, J. Math. Anal. Appl. 449 (2017), 531-552. https://doi.org/10.1016/j.jmaa.2016.12.008
  4. A. Azzollini, A. Pomponio, P. D'Avenia, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. Henri Poincaré Anal. Non Linéaire 27 (2010), 779-791. https://doi.org/10.1016/j.anihpc.2009.11.012
  5. S. Bai, Y. Song, D.D. Dušan, On \(p\)-Laplacian Kirchhoff-Schrödinger-Poisson type systems with critical growth on the Heisenberg group, Electron. Res. Arch. 31 (2023), 5749-5765. https://doi.org/10.3934/era.2023292
  6. V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), 283-293. https://doi.org/10.12775/tmna.1998.019
  7. J.M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier 19 (1969), 277-304. https://doi.org/10.5802/aif.319
  8. L. Capogna, D. Danielli, N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), 1765-1794. https://doi.org/10.1080/03605309308820992
  9. P. Cartier, Quantum mechanical commutation relations and theta functions, Proc. Sympos. Pure Math. 9 (1966), 361-383. https://doi.org/10.1090/pspum/009/0216825
  10. S. Chen, L. Lin, V.D. Rădulescu, Ground state solutions of the non-autonomous Schrödinger-Bopp-Podolsky system, Anal. Math. Phys. 12 (2022), 17. https://doi.org/10.22541/au.160252383.37333229/v1
  11. L. Cherfils, Y. Ilýasov, On the stationary solutions of generalized reaction diffusion equations with \(p,q\)-Laplacian, Pure Appl. Anal. 4 (2005), 9-22. https://doi.org/10.3934/cpaa.2005.4.9
  12. X. Dou, X. He, V.D. Rădulescu, Multiplicity of positive solutions for the fractional Schrödinger-Poisson system with critical nonlocal term, Bull. Math. Sci. 14 (2024), 2350012. https://doi.org/10.1142/s1664360723500121
  13. Z. Guo, Q. Shi, A Schrödinger-Poisson system with the critical growth on the first Heisenberg group, J. Contemp. Math. Anal. 58 (2023), 196-207. https://doi.org/10.3103/s1068362323030056
  14. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. https://doi.org/10.1007/bf02392081
  15. R. Illner, P.F. Zweifel, H. Lange, Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner-Poisson and Schrödinger-Poisson systems, Math. Methods Appl. Sci. 17 (1994), 349-376. https://doi.org/10.1002/mma.1670170504
  16. 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. https://doi.org/10.1090/s0894-0347-1988-0924699-9
  17. G.P. Leonardi, S. Masnou, On the isoperimetric problem in the Heisenberg group \(\mathbb{H}^{n}\), Ann. Mat. Pura Appl. 184 (2005), 533-553. https://doi.org/10.1007/s10231-004-0127-3
  18. S. Liang, P. Pucci, Y. Song, X. Sun, On a critical Choquard-Kirchhoff \(p\)-sub-Laplacian equation in \(\mathbb{H}^{n}\), Anal. Geom. Metr. Spaces 12 (2024): 20240006. https://doi.org/10.1515/agms-2024-0006
  19. S. Liang, P. Pucci, X. Sun, Normalized solutions for critical Schrödinger-Poisson system involving the \(p\)-subLaplacian in the Heisenberg group, Appl. Math. Lett. 158 (2024), 109-245. https://doi.org/10.1016/j.aml.2024.109245
  20. S. Liang, P. Pucci, X. Sun, Bifurcation problems for double critical Schrödinger-Poisson systems, Bull. Math. Sci. 2025 (2025), 2550029. https://doi.org/10.1142/s1664360725500298
  21. S. Liang, P. Pucci, X. Sun, Schrödinger-Poisson systems with the double critical growth on the first Heisenberg group, Appl. Math. Lett. 172 (2026), 109696. https://doi.org/10.1016/j.aml.2025.109696
  22. S. Liang, X. Sun, Sh. Liang, V.D. Rădulescu, Normalized solutions of the critical Schrödinger-Bopp-Podolsky system with logarithmic nonlinearity, Trans. London Math. Soc. 12 (2025), e70021. https://doi.org/10.1112/tlm3.70021
  23. S. Liang, X. Zheng, L. Guo, Normalized solutions for critical Choquard equations involving logarithmic nonlinearity in the Heisenberg group, Math. Methods Appl. Sci. 48 (2025), 3966-3978. https://doi.org/10.1002/mma.10528
  24. Sh. Liang, J. Ma, S. Shao, Multiple normalized solutions for Choquard equation involving the biharmonic operator and competing potentials in \(\mathbb{R}^N\), Bull. Math. Sci. 15 (2025), 2450017. https://doi.org/10.1142/s1664360724500176
  25. H. Liu, Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal. 32 (2016), 198-212. https://doi.org/10.1016/j.nonrwa.2016.04.007
  26. K. Liu, X. He, V.D. Rădulescu, Solutions with prescribed mass for the \(p\)-Laplacian Schrödinger-Poisson system with critical growth, J. Differential Equations 444 (2025), 113570. https://doi.org/10.1016/j.jde.2025.113570
  27. A. Loiudice, Improved Sobolev inequalities on the Heisenberg group, Nonlinear Anal. 62 (2005), 953-962. https://doi.org/10.1016/j.na.2005.04.015
  28. F. Nier, A stationary Schrödinger-Poisson systems arising from the modelling of electronic devices, Forum Math. 2 (1990), 489-510. https://doi.org/10.1515/form.1990.2.489
  29. P. Pucci, Critical Schrödinger-Hardy systems in the Heisenberg group, Discrete Contin. Dyn. Syst. Ser. S 12 (2019), 375-400. https://doi.org/10.3934/dcdss.2019025
  30. P. Pucci, L. Temperini, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional \(p\)-Laplacian in \(\mathbb{R}^{N}\), Calc. Var. Partial Differetial Equations 54 (2015), 2785-2806. https://doi.org/10.1007/s00526-015-0883-5
  31. P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBME Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986. https://doi.org/10.1090/cbms/065
  32. D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), 655-674. https://doi.org/10.1016/j.jfa.2006.04.005
  33. D. Ruiz, On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial andnonradial cases, Arch. Ration. Mech. Anal. 198 (2010), 349-368. https://doi.org/10.1007/s00205-010-0299-5
  34. X. Sun, Y. Song, S. Liang, On the critical Choquard-Kirchhoff problem on the Heisenberg group, Adv. Nonlinear Anal. 12 (2023), 210-236. https://doi.org/10.1515/anona-2022-0270
  35. W. Wei, X. Wu, A multiplicity result for quasilinear elliptic equations involving critical sobolev exponents, Nonlinear Anal. 18 (1992), 559-567. https://doi.org/10.1016/0362-546x(92)90210-6
  36. X. Zheng, L. Guo, S. Liang, Existence of positive solution for critical Schrödinger-Poisson system on the first Heisenberg group, Electron. Res. Arch. 33 (2025), 7052-7064. https://doi.org/10.3934/era.2025311
  • Xueyan Ma
  • College of Mathematics, Changchun Normal University, Changchun 130032, P.R. China
  • Shaoyun Shi
  • School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun 130022, P.R. China
  • Yueqiang Song (corresponding author)
  • College of Mathematics, Changchun Normal University, Changchun 130032, P.R. China
  • Communicated by Vicenţiu D. Rădulescu.
  • Received: 2026-04-05.
  • Revised: 2026-04-12.
  • Accepted: 2026-05-21.
  • Published online: 2026-06-08.
Opuscula Mathematica - cover

We advise that this website uses cookies to help us understand how the site is used. All data is anonymized. Recent versions of popular browsers provide users with control over cookies, allowing them to set their preferences to accept or reject all cookies or specific ones.