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Critical singular problems in Carnot groups

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Pith's one-line read Singular critical problems in Carnot groups have a sharp threshold: two positive solutions below it, one at it, none above it.

desk verdict A serious and mostly sound adaptation of the Euclidean singular-critical machinery to Carnot groups, but the proof of the second solution has a real gap for homogeneous dimension 5. read the letter →

arxiv 2506.07521 v1 pith:X4V7XEZU submitted 2025-06-09 math.AP

Stefano Biagi , Mattia Galeotti , Eugenio Vecchi This is my paper
classification math.AP MSC 35R0335B3335B2535J70
keywords PDEsonCarnotgroupsSingularequationsCriticalmildperturbationsub-LaplacianAubin-TalentifunctionsweaksolutionmultiplicitySobolevexponent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a sharp multiplicity threshold for a singular critical Dirichlet problem set in a bounded domain of a Carnot group, a non-Euclidean space whose natural Laplace operator is only degenerate-elliptic. The equation is $-\Delta_G u = \lambda u^{-\gamma} + u^{2^\star_Q-1}$ with $\gamma\in(0,1)$, and the critical exponent is the one coming from the Sobolev embedding of the Folland–Stein space. The main theorem says that a positive constant $\Lambda$ exists such that the problem has at least two positive weak solutions when $0<\lambda<\Lambda$, at least one when $\lambda=\Lambda$, and none when $\lambda>\Lambda$. This is the first power-type mild singular perturbation result of this kind on Carnot groups, and it shows that the classical Euclidean existence scheme survives even where the extremals of the Sobolev inequality are not explicitly known. If the theorem is right, variational Perron methods together with refined asymptotics of cut-off Sobolev extremals are sufficient machinery for this class of critical degenerate-elliptic problems.

What carries the argument

The load-bearing object is the family $U_{\varepsilon,a}(g)=\varphi(a^{-1}\diamond g)\,T_\varepsilon(a^{-1}\diamond g)$, where $T$ is a positive Sobolev extremal for the critical equation and $T_\varepsilon(g)=\varepsilon^{(2-Q)/2}T(\delta_{1/\varepsilon}(g))$ is its dilation rescaling; these functions play the role of the classical Euclidean Aubin–Talenti bubbles. Their job is to build test paths $u_\lambda+tRU_{\varepsilon,a}$ whose maximum energy lies strictly below $I_\lambda(u_\lambda)+S_G^{Q/2}/Q$, which prevents the loss of compactness and forces a second critical point. Three inputs carry the argument: the Sobolev-norm asymptotics $\|\nabla_G U_{\varepsilon}\|^2=S_G^{Q/2}+O(\varepsilon^{Q-2})$ and $\|U_{\varepsilon}\|_{2^\star_Q}^{2^\star_Q}=S_G^{Q/2}+O(\varepsilon^Q)$ imported from the existing theory of Sobolev extremals on Carnot groups; the convolution asymptotics $\int_\Omega u\,U_{\varepsilon,a}^p = K\varepsilon^{(Q-2)/2}+o(\varepsilon^{(Q-2)/2})$ for almost every $a$, obtained through the homogeneous-group convolution estimates; and a pointwise algebraic expansion for $|u+tRU_\varepsilon|^{2^\star_Q}$ supplied by the classical Brézis–Nirenberg lemma. The first solution, by contrast, is carried by the sub/supersolution (Perron) scheme, with the threshold $\Lambda$ bounded above by the principal-eigenvalue comparison $\mu_1\int_\Omega u e_1=\int_\Omega(\lambda u^{-\gamma}+u^{2^\star_Q-1})e_1$.

What would settle it

Fix a concrete Carnot group such as the Heisenberg group $\mathbb{H}^1$ and compute, for a numerically obtained Sobolev extremal $T$, the three asymptotics used in Lemma 4.3: $\|U_\varepsilon\|_{2^\star_Q}^{2^\star_Q}=S_G^{Q/2}+O(\varepsilon^Q)$, $\int_\Omega u U_{\varepsilon,a}^p=K\varepsilon^{(Q-2)/2}+o(\varepsilon^{(Q-2)/2})$, and the energy inequality $I_\lambda(u_\lambda+tR_0U_{\varepsilon,a})<I_\lambda(u_\lambda)+S_G^{Q/2}/Q$ for all $t\in[0,1]$. A single $\varepsilon_0>0$ for which any of these fails, while the hypotheses of the theorem hold, would falsify the central construction.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is that for every bounded domain with smooth enough boundary in any Carnot group and every $\gamma\in(0,1)$ there exists $\Lambda\in(0,\infty)$ such that the problem $-\Delta_G u=\lambda u^{-\gamma}+u^{2^\star_Q-1}$ with $u>0$ in $\Omega$ and $u=0$ on $\partial\Omega$ has at least two positive weak solutions for $0<\lambda<\Lambda$, at least one for $\lambda=\Lambda$, and none for $\lambda>\Lambda$. The first solution is obtained as a minimizer of the energy functional lying between an ordered subsolution and supersolution pair: the subsolution is the unique solution of the purely singular equation $-\Delta_G w=\lambda w^{-\gamma}$, and the supersolution is a solution of the same problem at a larger parameter $\lambda'$. The second solution comes from an Ekeland-type variational principle on the closed set of functions lying above the first solution, where the cut-off rescaled Sobolev extremals $U_{\varepsilon,a}$ are used to push the energy strictly below the critical compactness threshold $I_\lambda(u_\lambda)+S_G^{Q/2}/Q$. Non-existence for $\lambda>\Lambda$ follows by testing the equation with the principal eigenfunction of $-\Delta_G$.

Load-bearing premise

The existence of the second solution rests on the assumption that the positive entire solutions $T$ of the critical equation in the Carnot group satisfy the decay bounds $T(g)\le M_1\min\{1,|g|_G^{2-Q}\}$ and $T(g)\ge M_2(1+|g|_G)^{2-Q}$, and that these bounds yield the asymptotic expansions (2.23)–(2.24); if those expansions fail or are not uniform over the full class of Carnot groups covered by the statement, the mountain-pass energy gap in Lemma 4.3 can collapse and the second solution can be lost.

Editorial extensions

If this is right

  • For every $0<\lambda<\Lambda$ the boundary-value problem has two distinct positive weak solutions; one is a local minimizer of the energy and the other lies strictly above it.
  • At the threshold $\lambda=\Lambda$ at least one positive weak solution still exists, while for $\lambda>\Lambda$ no weak solution exists at all.
  • This is the first existence–multiplicity–non-existence trichotomy for power-type mild singular perturbations of critical semilinear equations in Carnot groups.
  • The second solution's construction needs only asymptotic expansions and decay bounds for Sobolev extremals, not their explicit closed form, so the argument is not tied to the Heisenberg group or other groups with known extremals.
  • The non-existence side has a clean form: once the parameter crosses the level where $\lambda t^{-\gamma}+t^{2^\star_Q-1}$ exceeds $\mu_1 t$ for every $t>0$, the principal-eigenvalue comparison rules out solutions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof suggests that $\Lambda$ is finite and positive for every Carnot group, but it does not compute $\Lambda$ explicitly; a natural next step would be to estimate it for concrete groups such as the Heisenberg group and to check numerically that the two solutions persist as $\lambda\to\Lambda$.
  • The same cut-off extremal argument may extend to strong singularities $\gamma\ge 1$, provided the singular energy term is handled by a different regularization, since the functional is no longer differentiable on the cone of positive functions.
  • Because the argument only uses decay and asymptotic expansions of Sobolev minimizers, one could try to replace the exact minimizer $T$ by a minimizing sequence with controlled decay, which might make the method applicable to $p\neq 2$ subelliptic Sobolev inequalities or to groups where minimizers are not explicitly known.
  • The two-case split in Section 4 (flat versus strict infimum on spheres) leaves open whether the second solution is always a mountain-pass point or can sometimes be a minimizer at a higher level; a stability analysis of the critical set could clarify the structure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies the Dirichlet problem (P)_λ on a bounded domain in a Carnot group, with a mild singular term λ u^{-γ} (0<γ<1) plus a critical nonlinearity u^{2*_Q-1}. Theorem 1.1 claims the existence of a threshold Λ>0 such that (P)_λ has at least two positive weak solutions for 0<λ<Λ, at least one for λ=Λ, and none for λ>Λ. The proof is structured as follows: the first solution is obtained by a variational Perron-type argument using the unique solution of the purely singular problem as subsolution and a solution of a nearby problem as supersolution; the threshold Λ is defined by a supremum and shown finite; the second solution is constructed by an Ekeland/mountain-pass argument on the set H_λ={u≥u_λ}, using rescaled extremal functions U_{ε,a} and their asymptotic expansions imported from the literature. The paper claims to be the first to treat power-type singular perturbations of critical problems in Carnot groups.

Significance. If the proof is completed, this is a natural and useful extension to the subelliptic Carnot setting of classical Euclidean results of Haitao and of Hirano–Saccon–Shioji. The paper is transparent about its external inputs: it imports extremal existence, decay bounds, and asymptotic expansions from [6,27,39] and adapts estimates from [11,30,55], with no fitted parameters or circular predictions. The variational strategy is standard but the adaptation to Carnot groups is nontrivial, and the paper carefully treats issues such as the strong maximum principle and the singular term. The main result, however, is currently not fully established for all Carnot groups because of a missing dimension case in the key energy estimate, as detailed below.

major comments (2)
  1. [§4.2, Lemma 4.3, Part (A3)] The case split in Part (A3) is incomplete: the text says 'First when q = p + 1 ≤ 3, which implies Q ≥ 5, and secondly the case q = 4, that gives Q = 4.' Since q = 2Q/(Q−2), the condition q ≤ 3 is equivalent to Q ≥ 6, not Q ≥ 5. The homogeneous dimension Q = 5 is therefore missing; it is realized, for example, by the Carnot group R × H_1, for which Q = 5 and p+1 = 10/3 > 3. For Q = 5, Lemma 4.4(2) (the case q > 3) applies, but the displayed estimates for X_ε and Y_ε in Case I are not justified, and no separate Q = 5 case is supplied. Consequently the energy bound (4.22), which is the basis of the mountain-pass construction of the second solution, is not established for such groups, and Theorem 1.1 as stated does not cover all Carnot groups. This is an internal gap in the proof of Lemma 4.3, not a problem with the imported expansions (2.18)–(2.24).
  2. [§3, proof of Lemma 3.4, after (3.14)] The closing display reads ∫_Ω ⟨∇_G u, ∇_G φ⟩ − ∫_Ω (λ u^{−γ} − u^{2*_Q−1}) φ ≥ o(1). The minus sign in front of u^{2*_Q−1} conflicts with (3.10) and with problem (P), for which the variational derivative contains +u^{2*_Q−1}. With the displayed sign, replacing φ by −φ does not yield the weak formulation. If this is a typo, it must be corrected; if it is not a typo, the Perron argument in Lemma 3.4 is not closed. Since Lemma 3.4 underpins the existence of the first solution, this point needs clarification and correction.
minor comments (5)
  1. [§2.1, Proposition 2.2] The statement says O ⊆ R^n, but the setting is Carnot groups; it should read O ⊆ G.
  2. [§4.2, Lemma 4.3, Part (D)] The lower bound for u on Ω^{in}_ε is justified by Corollary 2.3 or Remark 2.6(3), not by 'Lemma 3.4' as written in the text.
  3. [§3, Lemma 3.6] The notations S_j and S_j for the two supports are typographically indistinguishable; using e.g. S_j^+ and S_j^- would improve readability.
  4. [Throughout] Several typos remain, including 'in addiction', 'opportune', 'Analougously', 'Lispchitz', and 'torough'; these should be corrected.
  5. [§2.1, Eq. (2.8)] The identity for the sub-Laplacian under dilations appears to have a missing equals sign before λ^2(Δ_G u)∘δ_λ.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the proof adapts external results and its only self-citation is a non-load-bearing literature pointer.

full rationale

The paper's central claim (Theorem 1.1) is not circular. The first solution is obtained by a Perron-type subsolution/supersolution scheme built on Theorem 2.7, which is proved inside the paper, and the second solution is obtained by a Tarantello-type mountain-pass construction around the first solution using the family U_epsilon. The decay and asymptotic estimates (2.18), (2.19), (2.23), and (2.24) are imported from external prior work [6], [27], and [39] as parameter-free results with hypotheses that do not include the present paper's conclusion; they are independent benchmarks, not outputs of this paper. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' own prior work is invoked to force the construction. The only self-citation, [2], appears in the introduction as an example of related work with different leading operators and is not used to justify any proof step. Lambda is defined as the supremum of solvable lambda, so the non-existence statement for lambda > Lambda is tautological only in the sense of any threshold definition; the existence assertions for lambda < Lambda and lambda = Lambda require the actual arguments in Lemmas 3.1, 3.5, and Section 4 and are not presupposed. The skeptic's concern about the case split Q >= 5 instead of Q >= 6 in Lemma 4.3(A3) is a possible internal correctness gap for homogeneous dimension Q = 5, not a circularity: if valid, it would mean the proof does not establish the second solution for all Carnot groups, not that the theorem's content is identical to its assumptions. Overall, the derivation is self-contained against external benchmarks and the circularity burden is near zero.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no fitted parameters and no new entities. The proof imports structural facts about Carnot groups: the Sobolev embedding, the existence of extremals with pointwise decay, the asymptotic expansions of cut-off extremals, and the weak Harnack inequality. These are external theorems with independent proofs, so they are domain assumptions rather than circular inputs. A few auxiliary choices (cut-off functions, base point a, radius R0) are not free parameters.

assumptions (5)
  • standard math Carnot group structure and Folland-Stein Sobolev inequality with compact embeddings below the critical exponent
    Used in Section 2.2 to define S^1_0(Omega), to obtain Sobolev and compact embeddings, and to control the critical term.
  • domain assumption Existence, positivity, and asymptotic properties of entire solutions T of the critical equation, specifically bounds (2.18)-(2.19) and asymptotics (2.23)-(2.24) from [6,27,39]
    These imported results are the Carnot analogue of the Aubin-Talenti functions and are essential for Lemma 4.3 and the mountain-pass estimates.
  • standard math Weak Harnack inequality and strong maximum principle for -Delta_G + c with bounded nonnegative c, inherited from the X-elliptic theory of Gutierrez-Lanconelli [29]
    The paper proves Corollary 2.3 and Proposition 2.4 from these results; positivity and comparison arguments depend on them.
  • domain assumption The domain Omega is bounded, connected, and has boundary smooth enough for Dirichlet test functions and compact Sobolev embeddings
    No explicit regularity beyond 'smooth enough' is stated; the proof uses C^infinity_0(Omega) test functions and the compact embedding S^1_0(Omega) into L^p for p<2*.
  • standard math Ekeland's variational principle and standard critical point theory on complete metric spaces
    Used in Section 4 to extract the approximating sequence v_k satisfying (4.28).

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Cite this review

Pith. "Pith review of Critical singular problems in Carnot groups." pith.science (2026). https://pith.science/paper/X4V7XEZU

@misc{pith2026250607521,
  author       = {Pith},
  title        = {Pith review of: Critical singular problems in Carnot groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X4V7XEZU}},
  note         = {Machine review of arXiv:2506.07521}
}
read the original abstract

We consider a power-type mild singular perturbation of a Dirichlet semilinear critical problem settled in an open and bounded set in a Carnot group. Here, the term critical has to be understood in the sense of the Sobolev embedding. We aim to prove the existence of two positive weak solutions: the first one is obtained by means of the variational Perron's method, while for the second one we adapt a classical argument relying on proper estimates of a family of functions which mimic the role of the classical Aubin-Talenti functions in the Euclidean setting. Our results fall in the framework of semilinear PDEs in Carnot group but, as far as we know, are the first ones dealing with singular perturbations of power-type.

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Works this paper leans on

57 extracted references · 57 canonical work pages

  1. [2]

    Biagi, E

    S. Biagi, E. Vecchi, Multiplicity of positive solutions for mixed local-nonlocal singular critical problems , Calc. Var. Partial Differ. Equ. 63, (2024), 221. 2

    work page 2024

  2. [1]

    Badiale, G

    M. Badiale, G. Tarantello, Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities, Nonlinear Anal. Theory Methods Appl. 29, (1997), 639–677. 3, 29, 39

    work page 1997

  3. [3]

    Birindelli, I

    I. Birindelli, I. Capuzzo Dolcetta, A. Cutr ` ı,Liouville theorems for semilinear equations on the Heisenberg group, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire14, (1997), 295–308. 2

    work page 1997

  4. [4]

    Birindelli, J.V

    I. Birindelli, J.V. Prajapat, Nonlinear Liouville theorems in the Heisenberg group via the moving plane method, Comm. Partial Differential Equations 24(9-10), (1999), 1875–1890. 2

    work page 1999

  5. [5]

    Bonfiglioli, E

    A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub- Laplacians. Springer Monographs in Mathematics, vol. 26. Springer, New York, NY (2007). 4

    work page 2007

  6. [6]

    Bonfiglioli, F

    A. Bonfiglioli, F. Uguzzoni, Nonlinear Liouville theorems for some critical problems on H-type groups , J. Funct. Anal. 207(1), (2004), 161–215. 2, 7

    work page 2004

  7. [7]

    Bony, Principe du maximum, in´ egalit´ e de Harnack et unicit´ e du probl` eme de Cauchy pour les op´ erateurs elliptiques d´ eg´ en´ er´ es, Ann

    J.M. Bony, Principe du maximum, in´ egalit´ e de Harnack et unicit´ e du probl` eme de Cauchy pour les op´ erateurs elliptiques d´ eg´ en´ er´ es, Ann. Inst. Fourier Grenobles 19(1), (1969), 277–304. 7

    work page 1969

  8. [8]

    Brandolini, M

    L. Brandolini, M. Rigoli, A.G. Setti, Positive solutions of Yamabe-type equations on the Heisenberg group , Duke Math. J. 91, (1998), 241–296. 2

    work page 1998

Show all 57 references
  1. [9]

    Br´ ezis, E

    H. Br´ ezis, E. Lieb, A relations between pointwise convergence of functions and convergence of integrals , Proc. Amer. Math. Soc. 88, (1983), 486–490. 17

    1983

  2. [10]

    Br´ ezis, L

    H. Br´ ezis, L. Nirenberg, Positive solutions of nonlinear elliptic equation involving the critical Sobolev exponent, Comm. Pure Appl. Math. 36, (1983), 437–477. 2

    1983

  3. [11]

    Br´ ezis, L

    H. Br´ ezis, L. Nirenberg, A minimization problem with critical exponent and nonzero data , in Symmetry in Nature (a volume in honor of L. Radicati), Scuola Normale Superiore Pisa, (1989), Volume I, 129–140. 3, 34, 36

    1989

  4. [12]

    Cheng, A

    J.-H. Cheng, A. Malchiodi, P. Yang, On the Sobolev quotient of three-dimensional CR manifolds , Rev. Mat. Iberoam. 39(6), (2023), 2017–2066. 2

    2023

  5. [13]

    Citti, Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian , Ann

    G. Citti, Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian , Ann. Mat. Pura. Appl. 169, (1995), 375–392. 2

    1995

  6. [14]

    Citti, F

    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. 2

    2001

  7. [15]

    Crandall, P.H

    M.G. Crandall, P.H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity , Comm. Partial Differential Equations 2, (1977), 193–222. 3

    1977

  8. [16]

    Ekeland, On the variational principle , J

    I. Ekeland, On the variational principle , J. Math. Anal. Appl. 47, (1974), 324–353. 27

    1974

  9. [17]

    Felli, F

    V. Felli, F. Uguzzoni, Some existence results for the Webster scalar curvature problem in presence of symmetry, Ann. Mat. Pura Appl. 183(4), (2004), 469–493. 2

    2004

  10. [18]

    Folland, Subelliptic estimates and function spaces on nilpotent Lie groups , Ark

    G.B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups , Ark. Mat. 13, (1975), 161–207. 2, 5, 36

    1975

  11. [19]

    Folland, The Heisenberg group and its relatives in the work of Elias M

    G.B. Folland, The Heisenberg group and its relatives in the work of Elias M. Stein , J. Geom. Anal. 31(7), (2021), 6681–6697. 2

    2021

  12. [20]

    Folland, E.M

    G.B. Folland, E.M. Stein, Hardy Spaces on Homogeneous Groups , Princeton University Press, Dec. 2020. 3, 35

    2020

  13. [21]

    Franchi, R

    B. Franchi, R. Serapioni, F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields , Boll. Un. Mat. Ital. B 11(7), (1997), 83–117. 6

    1997

  14. [22]

    Gamara, The CR Yamabe conjecture the case n = 1, J

    N. Gamara, The CR Yamabe conjecture the case n = 1, J. Eur. Math. Soc. 3(2), (2001), 105–137. 2

    2001

  15. [23]

    Gamara, R

    N. Gamara, R. Yacoub, CR Yamabe conjecture the conformally flat case , Pacific J. Math. 201(1), (2001), 121–175. 2

    2001

  16. [24]

    Garagnani, F

    E. Garagnani, F. Uguzzoni, A multiplicity result for a degenerate elliptic equation with critical growth on noncontractible domains, Topol. Methods Nonlinear Anal. 22(1), (2003), 53–68. 2

    2003

  17. [25]

    Garofalo, E

    N. Garofalo, E. Lanconelli, Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41(1), (1992), 71–98. 2

    1992

  18. [26]

    Garofalo, D

    N. Garofalo, 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), 453—516. 2

    2000

  19. [27]

    Garofalo, D

    N. Garofalo, D. Vassilev, Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type , Duke Math. J. 106(3), (2001), 411–448. 2, 7

    2001

  20. [28]

    Giacomoni, T

    J. Giacomoni, T. Mukherjee, K. Sreenadh, A global multiplicity result for a very singular critical nonlocal equation, Topol. Methods Nonlinear Anal. 54 (2019), 345–370. 2 CRITICAL SINGULAR PROBLEMS IN CARNOT GROUPS 45

    2019

  21. [29]

    Guti´ errez, E

    C.E. Guti´ errez, E. Lanconelli, Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for X-elliptic operators , Comm. Partial Differential Equations 28(11-12), (2003), 1833–1862. 8, 9, 12

    2003

  22. [30]

    Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J

    Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations 189(2), (2003), 487–512. 2, 3, 17, 18, 19, 20, 21, 22, 23, 24, 27, 29

    2003

  23. [31]

    Hirano, C

    N. Hirano, C. Saccon, N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities , Adv. Differential Equations 9(1-2), (2004), 197–220. 2

    2004

  24. [32]

    H¨ ormander,Hypoelliptic second order differential equations , Acta Math

    L. H¨ ormander,Hypoelliptic second order differential equations , Acta Math. 119, (1967), 147–171. 2

    1967

  25. [33]

    Jerison, J

    D. Jerison, J. Lee, The Yamabe problem on CR manifolds , J. Differential Geom. 25(2), (1987), 167–197. 2

    1987

  26. [34]

    Jerison, J

    D. Jerison, J. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1(1), (1988), 1–13. 2

    1988

  27. [35]

    Jerison, J

    D. Jerison, J. Lee, Intrinsic CR normal coordinates and the CR Yamabe problem , J. Differ. Geom. 29, (1989), 303–343. 2

    1989

  28. [36]

    Kinderlehrer, G

    D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Volume 88 of Pure and Applied Mathematics. Academic Press, Inc. Harcourt Brace Jovanovich, Publishers, New York-London, 1980. 15

    1980

  29. [37]

    Kumar, V.D

    D. Kumar, V.D. Radulescu, K. Sreenadh, Singular elliptic problems with unbalanced growth and critical exponent, Nonlinearity 33(7), (2020), 3336–3369. 2

    2020

  30. [38]

    Lanconelli, F

    E. Lanconelli, F. Uguzzoni, Non-existence results for semilinear Kohn–Laplace equations in unbounded domains, Comm. Partial Differential Equations 25, (2000), 1703–1739. 2

    2000

  31. [39]

    Loiudice, Semilinear subelliptic problems with critical growth on Carnot groups , Manuscripta Math

    A. Loiudice, Semilinear subelliptic problems with critical growth on Carnot groups , Manuscripta Math. 124, (2007), 247–259. 2, 3, 7, 8

    2007

  32. [40]

    Loiudice, Critical growth problems with singular nonlinearities on Carnot groups , Nonlinear Anal

    A. Loiudice, Critical growth problems with singular nonlinearities on Carnot groups , Nonlinear Anal. 126, (2015), 415–436. 3

    2015

  33. [41]

    Loiudice, Optimal decay of p-Sobolev extremals on Carnot groups, J

    A. Loiudice, Optimal decay of p-Sobolev extremals on Carnot groups, J. Math. Anal. Appl. 470(1), (2019), 619–631. 2

    2019

  34. [42]

    Loiudice, Critical problems with Hardy potential on stratified Lie groups , Adv

    A. Loiudice, Critical problems with Hardy potential on stratified Lie groups , Adv. Differential Equations 28(1-2), (2023), 1–33. 3

    2023

  35. [43]

    G. Lu, J. Wei, On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups, Electron. Res. Announc. Amer. Math. Soc. 3, (1997), 83–89. 2

    1997

  36. [44]

    Maalaoui, V

    A. Maalaoui, V. Martino, Multiplicity result for a nonhomogeneous Yamabe type equation involving the Kohn Laplacian, J. Math. Anal. Appl. 399(1), (2013), 333–339. 2

    2013

  37. [45]

    Maalaoui, V

    A. Maalaoui, V. Martino, A. Pistoia, Concentrating solutions for a sub-critical sub-elliptic problem , Dif- ferential Integral Equations 26(11-12), (2013), 1263–1274. 2

    2013

  38. [46]

    Maalaoui, V

    A. Maalaoui, V. Martino, G. Tralli, Complex group actions on the sphere and sign changing solutions for the CR-Yamabe equation, J. Math. Anal. Appl. 431(1), (2015), 126–135. 2

    2015

  39. [47]

    Malchiodi, F

    A. Malchiodi, F. Uguzzoni, A perturbation result for the Webster scalar curvature problem on the CR sphere, J. Math. Pures Appl. 81(10), (2002), 983–997. 2

    2002

  40. [48]

    Molica Bisci, D

    G. Molica Bisci, D. Repovs, Yamabe-type equations on Carnot groups , Potential Anal. 46(2), (2017), 369–383. 2

    2017

  41. [49]

    Oliva, F

    F. Oliva, F. Petitta, Singular Elliptic PDEs: an extensive overview , Partial Differ. Equ. Appl. 6, (2025), Article 6. 3

    2025

  42. [50]

    Palatucci, M

    G. Palatucci, M. Piccinini, L. Temperini, Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group , Adv. Calc. Var. (2024), https://doi.org/10.1515/ acv-2024-0044 2

    2024

  43. [51]

    Rothschild, E.M

    L.P. Rothschild, E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137(3-4), (1976), 247–320. 2

    1976

  44. [52]

    Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A-weights, Rev

    S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A-weights, Rev. Mat. Iberoam. 12(2), (1996), 337–410. 4

    1996

  45. [53]

    Struwe, Variational Methods, Springer-Verlag, Berlin, 1990

    M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990. xiv+244 pp. 3

    1990

  46. [54]

    Sun, S.P

    Y.J. Sun, S.P. Wu, Y.M. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems , J. Differential Equations 176, (2001), 511–531. 12

    2001

  47. [55]

    Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent , Ann

    G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent , Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire9, (1992), 281–304. 3, 32, 40

    1992

  48. [56]

    Uguzzoni, A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group , NoDEA Nonlinear Differential Equations Appl

    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), 191–206. 2 46 S. BIAGI, M. GALEOTTI, AND E. VECCHI

    1999

  49. [57]

    Uguzzoni, A note on Yamabe-type equations on the Heisenberg group , Hiroshima Math

    F. Uguzzoni, A note on Yamabe-type equations on the Heisenberg group , Hiroshima Math. J. 30, (2000), 179–189. 2 (S. Biagi) Dipartimento di Matematica Politecnico di Milano Via Bonardi 9, 20133 Milano, Italy Email address : stefano.biagi@polimi.it (M. Galeotti) Dipartimento di...

    2000

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