Opuscula Mathematica
Opuscula Math. 46, no. 5 (2026), 599-622

Multiplicity results for subelliptic problems with Hardy potential on stratified Lie groups

Yergen Aikyn
Aidyn Kassymov
Michael Ruzhansky
Abstract. In this paper, we study the existence and multiplicity of solutions to the following subelliptic problem with Hardy potential \[\begin{cases}-\Delta_{\mathbb{G}} u(\xi)-\mu \frac{u(\xi)}{|x^{\prime}|^{2}}=u^{q}(\xi)+\lambda u^{r}(\xi)&\text{in}\; \Omega , \\ u(\xi)\gt 0 &\text{in}\; \Omega, \\ u(\xi)=0 &\text{on}\; \partial \Omega,\end{cases}\] where \(\Omega \subset \mathbb{G}\) is a bounded domain containing the origin with smooth boundary \(\partial \Omega\), \(0\lt\mu\lt\bar{\mu}:=\left(\frac{N-2}{2}\right)^{2}\) with \(N\gt 2\), and \(\lambda\gt 0\). Here, the Hardy potential is defined on the first stratum of a stratified Lie group \(\mathbb{G}\). Specifically, we have \(|x^{\prime}|=\sqrt{x_{1}^{2}+\ldots+x_{N}^{2}},\) \(x^{\prime}=(x_{1},\ldots, x_{N}) \in \mathbb{R}^{N}\), where \(N\) is the dimension of the first stratum of the stratified Lie group \(\mathbb{G}\). Under the assumptions \(1\lt r+1\lt 2\lt q+1\lt 2^*:=\frac{2Q}{Q-2}\), we prove the existence of at least two positive solutions provided that \(\lambda\) is sufficiently small.
Keywords: subelliptic problem, Hardy potential, Palais-Smale condition, concentration-compactness principle.
Mathematics Subject Classification: 35R03, 35A15, 35H20.
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References ▼
  1. [1] B. Abdellaoui, I. Peral, Some results for semilinear elliptic equations with critical potential, Proc. R. Soc. Edinb., Sect. A. Math. 132 (2002), 1-24. https://doi.org/10.1017/s0308210500001505
  2. [2] A. Ambrosetti, P. Rabinowitz, Dual methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
  3. [3] J.G. Azorero, I.P. Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 877-895. https://doi.org/10.1090/s0002-9947-1991-1083144-2
  4. [4] B. Barrios, M. Medina, I. Peral, Some remarks on the solvability of non-local elliptic problems with the Hardy potential, Commun. Contemp. Math. 16 (2014), 1350046. https://doi.org/10.1142/s0219199713500466
  5. [5] T. Bhattacharya, A. Mohammed, On a strong maximum principle for fully nonlinear subelliptic equations with Hörmander condition, Calc. Var. Partial Differential Equations 60 (2021), Paper No. 9.
  6. [6] A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer, Berlin, 2007. https://doi.org/10.1007/978-3-540-71897-0
  7. [7] S. Bordoni, R. Filippucci, P. Pucci, Existence problems on Heisenberg groups involving Hardy and critical terms, J. Geom. Anal. 30 (2020), 1887-1917.
  8. [8] H. Brezis, L. Dupaigne, A. Tesei, On a semilinear elliptic equation with inverse-square potential, Selecta Math. (N.S.) 11 (2005), Paper No. 1. https://doi.org/10.1007/s00029-005-0003-z
  9. [9] L. Capogna, D. Danielli, S.D. Pauls, J.T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics, vol. 259, Birkhäuser Verlag, Basel, 2007. https://doi.org/10.1007/978-3-7643-8133-2
  10. [10] H. Chen, A. Quaas, F. Zhou, On nonhomogeneous elliptic equations with the Hardy-Leray potentials, J. Anal. Math. 144 (2021), 305-334. https://doi.org/10.1007/s11854-021-0182-3
  11. [11] J. Chen, Existence of solutions for a nonlinear PDE with an inverse square potential, J. Differ. Equ. 195 (2003), 497-519. https://doi.org/10.1016/s0022-0396(03)00093-7
  12. [12] J. Chen, E.M. Rocha, Existence of solution of subelliptic equations on the Heisenberg group with critical growth and double singularities, Opuscula Math. 33 (2013), 237-254. https://doi.org/10.7494/opmath.2013.33.2.237
  13. [13] G. Citti, Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian, Ann. Mat. Pura Appl. 169 (1995), 375-392. https://doi.org/10.1007/bf01759361
  14. [14] G. Citti, F. Uguzzoni, Critical semilinear equations on the Heisenberg group: the effect of the topology of the domain, Nonlinear Anal. 46 (2001), 399-417. https://doi.org/10.1016/s0362-546x(00)00138-3
  15. [15] L. D’Ambrosio, Some Hardy inequalities on the Heisenberg group, Differ. Equ. 40 (2004), 552-564.
  16. [16] L. D’Ambrosio, Hardy-type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 451-486. https://doi.org/10.2422/2036-2145.2005.3.04
  17. [17] L. Dupaigne, A nonlinear elliptic PDE with the inverse square potential, J. Anal. Math. 86 (2002), 359-398.
  18. [18] M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential, Nonlinear Anal. 193 (2020), 111311. https://doi.org/10.1016/j.na.2018.07.008
  19. [19] A. Ferrero, F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differ. Equ. 177 (2001), 494-522. https://doi.org/10.1006/jdeq.2000.3999
  20. [20] V. Fischer, M. Ruzhansky, Quantization on Nilpotent Lie Groups, Progress in Mathematics, vol. 314, Birkhäuser, 2016. https://doi.org/10.1007/978-3-319-29558-9_2
  21. [21] N. Garofalo, E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble) 40 (1990), 313-356. https://doi.org/10.5802/aif.1215
  22. [22] N. Garofalo, E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41 (1992), 71-98. https://doi.org/10.1512/iumj.1992.41.41005
  23. [23] 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. https://doi.org/10.1002/(sici)1097-0312(199610)49:10<1081::aid-cpa3>3.0.co;2-a
  24. [24] N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Am. Math. Soc. 12 (2000), 5703-5743. https://doi.org/10.1090/s0002-9947-00-02560-5
  25. [25] D.S. Jerison, J.M. Lee, Extremals of the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1-13. https://doi.org/10.2307/1990964
  26. [26] A. Kassymov, D. Suragan, Multiplicity of positive solutions for a nonlinear equation with a Hardy potential on the Heisenberg group, Bull. Sci. Math. 165 (2020), 102916. https://doi.org/10.1016/j.bulsci.2020.102916
  27. [27] A. Loiudice, Critical problems with Hardy potential on Stratified Lie Groups, Adv. Differential Equations 28 (2023), 1-33. https://doi.org/10.57262/ade028-0102-1
  28. [28] H. Mokrani, Semilinear subelliptic equations on the Heisenberg group with a singular potential, Comm. Pure Appl. Anal. 8 (2009), 1619-1636. https://doi.org/10.3934/cpaa.2009.8.1619
  29. [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. https://doi.org/10.1090/s0002-9939-01-06011-7
  30. [30] P. Pucci, L. Temperini, On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), Paper No. 007.
  31. [31] M. Ruzhansky, D. Suragan, On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and \(p\)-sub-Laplacian inequalities on stratified groups, J. Differential Equations 262 (2017), 1799-1821. https://doi.org/10.1016/j.jde.2016.10.028
  32. [32] M. Ruzhansky, D. Suragan, Hardy inequalities on homogeneous groups, Progress in Math., vol. 327, Birkhäuser, 2019, 71-127. https://doi.org/10.1007/978-3-030-02895-4_3
  33. [33] M. Ruzhansky, D. Suragan, Green's identities, comparison principle and uniqueness of positive solutions for nonlinear \(p\)-sub-Laplacian equations on stratified Lie groups, Potential Anal. 53 (2020), 645-658. https://doi.org/10.1007/s11118-019-09782-y
  34. [34] M. Ruzhansky, B. Sabitbek, D. Suragan, Weighted \(L^p\)-Hardy and \(L^p\)-Rellich inequalities with boundary terms on stratified Lie groups, Rev. Math. Complutense 32 (2019), 19-35. https://doi.org/10.1007/s13163-018-0268-3
  35. [35] P. Sintzoff, Existence of minimizer of some functionals involving Hardy-type inequalities, Topol. Methods Nonlinear Anal. 28 (2006), 105-115.
  36. [36] D. Smets, A concentration-compactness lemma with applications to singular eigenvalue problems, J. Funct. Anal. 167 (1999), 463-480.
  37. [37] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990. https://doi.org/10.1002/zamm.19930730111
  38. [38] S.K. Vodop'yanov, Weighted Sobolev spaces and the boundary behavior of solutions of degenerate hypoelliptic equations, Siberian Math. J. 36 (1995), 246-264 (translated from Sibirsk. Mat. Zh. 36 (1995), 278-300). https://doi.org/10.1007/bf02110147
  39. [39] M. Willem, A decomposition lemma and critical minimization problems, [in:] Proceedings of the Workshop on Morse Theory, Minimax Theory and Their Applications, Chinese Academy of Sciences, Beijing, 2003, 247-256.
About the authors ▼
Yergen Aikyn (corresponding author)
https://orcid.org/0009-0003-9811-0856
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
Aidyn Kassymov
https://orcid.org/0000-0003-0080-4503
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan; Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
Michael Ruzhansky
https://orcid.org/0000-0001-8633-5570
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium; School of Mathematical Sciences, Queen Mary University of London, United Kingdom
About the article ▼
Communicated by Daniele Cassani.
Received: 2026-01-30.
Revised: 2026-09-21.
Accepted: 2026-10-08.
Published online: 2026-10-28.
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Yergen Aikyn, Aidyn Kassymov, Michael Ruzhansky, Multiplicity results for subelliptic problems with Hardy potential on stratified Lie groups, Opuscula Math. 46, no. 5 (2026), 599-622, https://doi.org/10.7494/OpMath.202610081.
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