pith. sign in

arxiv: 2605.25157 · v1 · pith:RJ46GD67new · submitted 2026-05-24 · 🧮 math.AP

Brezis-Nirenberg problems for mixed local-nonlocal operators with superlinear perturbations: compactness and applications

Mousomi Bhakta , Nirjan Biswas , Paramananda Das This is my paper

Pith reviewed 2026-06-29 23:34 UTC · model grok-4.3

classification 🧮 math.AP
keywords Brezis-Nirenberg problemmixed local-nonlocal operatorcompactnessconcentration-compactnesssign-changing solutionscritical exponentsubcritical approximation
0
0 comments X

The pith

Bounded sequences of solutions to subcritical mixed Brezis-Nirenberg problems converge strongly to nontrivial solutions of the critical problem under specific ranges of p and N.

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

The paper establishes a compactness result showing that for p in (2 + 4s/(N-2), 2*) when N > 6-4s and for p in (2*-1, 2*) when N ≤ 6-4s, any bounded sequence of solutions to the subcritical problems converges strongly in the energy space to a nontrivial solution of the critical mixed local-nonlocal equation. This compactness is proved by adapting concentration-compactness techniques to the non-homogeneous operator, which is the first such result for this type of operator. The result then yields the existence of infinitely many sign-changing solutions to the critical problem. A reader would care because the compactness removes a common obstruction to existence proofs in critical exponent problems involving mixed operators.

Core claim

For the mixed local-nonlocal Brezis-Nirenberg problem -Δu + (-Δ)^s u = λ |u|^{p-2}u + |u|^{2*-2}u in Ω with exterior Dirichlet condition, any bounded sequence of solutions to the approximating subcritical problems with exponents p_n → 2* is relatively compact in the energy space and converges strongly to a nontrivial solution of the critical problem, when p lies in (2 + 4s/(N-2), 2*) for N > 6-4s and in (2*-1, 2*) for N ≤ 6-4s.

What carries the argument

The adapted concentration-compactness argument for the mixed local-nonlocal operator, obtained by non-trivial modification of the methods of Devillanova-Solimini and Yan-Yang-Yu.

If this is right

  • The critical mixed problem admits infinitely many sign-changing solutions under the same conditions on N and p.
  • The compactness holds for the non-homogeneous operator without additional assumptions on Ω.
  • The same ranges of p and N suffice for both the compactness and the multiplicity result.
  • The adapted proof technique applies to this class of mixed operators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The compactness result may extend to other superlinear perturbations or different combinations of local and nonlocal terms.
  • Numerical checks in low dimensions could test whether the boundary values of p and N are sharp.
  • Similar compactness statements could be sought for variable-order or higher-order mixed operators.

Load-bearing premise

The concentration-compactness methods can be adapted directly to the mixed operator without extra conditions on the domain or further restrictions on the bounded sequences.

What would settle it

A bounded sequence of solutions to the subcritical problems with p_n → 2* that fails to converge strongly in the energy space to any solution of the critical problem.

read the original abstract

In this paper, we consider the following mixed local nonlocal Brezis-Nirenberg problem \begin{equation}\label{crit_pro_abstract}\tag{$\mathcal{P}_{2^*}$} -\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{2^*-2}u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^N \setminus \Omega, \end{equation} where $\Omega\subset\mathbb{R}^N$ is a bounded domain, $N\geq3$, $s\in(0,1)$, $\lambda>0$, and $2\leq p<2^*=\frac{2N}{N-2}$. We establish a compactness result for the following class of subcritical/critical problems \begin{equation}\label{sub_pro_abstract}\tag{$\mathcal{P}_{p_n}$} -\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{p_n-2}u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^N \setminus \Omega, \end{equation} where $p_n \in (p,2^* ]$ and $p_n\to 2^*$. Specifically, for $p \in (2+\frac{4s}{N-2},2^*)$ when $N>6-4s$, and for $p \in (2^*-1,2^*)$ when $N\leq6-4s$, we prove that any bounded sequence of solutions $\{u_n\}$ to \eqref{sub_pro_abstract} is relatively compact in the energy space, and converges strongly to a nontrivial solution to \eqref{crit_pro_abstract}. This is the first paper to address this type of compactness result for a non-homogeneous operator. Due to the presence of the non-homogeneous operator, our proof requires a non-trivial adaptation of the methods developed by Devillanova and Solimini (Adv. Differential Equations, 2002) and Yan, Yang, and Yu (J. Funct. Anal., 2015). As an application of this compactness result, under the same ranges of $N$ and $p$, we prove that \eqref{crit_pro_abstract} admits infinitely many sign-changing solutions. We anticipate that our methodology will be applicable to a broader class of related problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes a compactness result for bounded sequences of solutions {u_n} to the subcritical problems (P_{p_n}) with p_n → 2*, showing relative compactness in the energy space and strong convergence to a nontrivial solution of the critical Brezis-Nirenberg problem (P_{2*}) for the mixed operator -Δu + (-Δ)^s u. The ranges are p ∈ (2 + 4s/(N-2), 2*) when N > 6-4s and p ∈ (2*-1, 2*) when N ≤ 6-4s. As an application, the critical problem is shown to admit infinitely many sign-changing solutions. The proof adapts concentration-compactness techniques from Devillanova-Solimini (2002) and Yan-Yang-Yu (2015) to handle the non-homogeneous operator.

Significance. If the compactness result holds, the work is significant as the first compactness theorem for this class of non-homogeneous mixed local-nonlocal operators. It successfully adapts profile decomposition, energy splitting, and ruling out of vanishing/dichotomy to the mixed setting while preserving the boundedness assumption, enabling a multiplicity result for sign-changing solutions. The methodology is positioned for broader applicability to related mixed-operator problems.

minor comments (3)
  1. [Introduction] The energy space (intersection of H^1_0(Ω) and H^s_0(Ω)) should be defined explicitly at the first appearance in the introduction rather than only via the abstract.
  2. [Introduction] The claim that this is the first such compactness result for a non-homogeneous operator would be strengthened by a short paragraph contrasting with prior works on homogeneous cases in the introduction.
  3. [§2] Notation for the sequence p_n and the limit 2* is clear in the abstract but should be restated once in §2 when the problems are introduced formally.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive report, which accurately summarizes the compactness result for bounded sequences solving the subcritical mixed local-nonlocal problems and its application to the existence of infinitely many sign-changing solutions for the critical problem. The referee correctly identifies the novelty in adapting profile decomposition and concentration-compactness arguments to the non-homogeneous mixed operator -Δ + (-Δ)^s. We appreciate the assessment of significance and the recommendation for minor revision. No specific major comments were listed in the report, so we address the overall evaluation below.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a compactness result for sequences of solutions to subcritical approximations converging to the critical Brezis-Nirenberg problem for the mixed local-nonlocal operator. The derivation explicitly adapts concentration-compactness techniques from the external references Devillanova-Solimini (2002) and Yan-Yang-Yu (2015), with stated modifications to handle the non-homogeneous operator, profile decomposition, and energy splitting while preserving the boundedness assumption. No load-bearing steps reduce by construction to self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior author work; the cited methods are independent and the result is presented as a direct (non-trivial) adaptation without internal reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard tools from functional analysis for elliptic PDEs; no free parameters or new entities are introduced.

axioms (2)
  • standard math Sobolev embeddings and concentration-compactness principles hold in the energy space for the mixed operator
    Implicit in the definition of the energy space and the statement of relative compactness.
  • domain assumption The methods of Devillanova-Solimini and Yan-Yang-Yu admit a non-trivial adaptation to the mixed local-nonlocal setting
    Stated directly in the abstract as required for the proof.

pith-pipeline@v0.9.1-grok · 5989 in / 1509 out tokens · 37563 ms · 2026-06-29T23:34:48.976060+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A priori bounds for energy-bounded solutions of critical polyharmonic equations

    math.AP 2026-05 unverdicted novelty 6.0

    Proves uniform a priori bounds and a new pointwise description for bounded-energy blowing-up solutions of critical polyharmonic equations in high dimensions via asymptotic analysis.

Reference graph

Works this paper leans on

36 extracted references · 26 canonical work pages · cited by 1 Pith paper

  1. [1]

    C. A. Antonini and M. Cozzi. Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type.J. Differential Equations, 425:342–382, 2025. ISSN 0022- 0396,1090-2732. doi: 10.1016/j.jde.2025.01.030. URLhttps://doi.org/10.1016/j.jde.2025. 01.030. 14, 15

    work page doi:10.1016/j.jde.2025.01.030 2025

  2. [2]

    F. V. Atkinson, H. Brezis, and L. A. Peletier. Solutions d’´ equations elliptiques avec exposant de Sobolev critique qui changent de signe.C. R. Acad. Sci. Paris S´ er. I Math., 306(16):711–714,

  3. [3]

    T. Aubin. Probl` emes isop´ erim´ etriques et espaces de Sobolev.J. Differential Geometry, 11(4): 573–598, 1976. ISSN 0022-040X,1945-743X. URLhttp://projecteuclid.org/euclid.jdg/ 1214433725. 10

    1976

  4. [4]

    Barrios, E

    B. Barrios, E. Colorado, R. Servadei, and F. Soria. A critical fractional equation with concave- convex power nonlinearities.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 32(4):875–900, 2015. ISSN 0294-1449,1873-1430. doi: 10.1016/j.anihpc.2014.04.003. URLhttps://doi.org/10.1016/ j.anihpc.2014.04.003. 5

    work page doi:10.1016/j.anihpc.2014.04.003 2015

  5. [5]

    Bhakta, N

    M. Bhakta, N. Biswas, and P. Das. Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions.Discrete and Continuous Dynamical Systems, 2026. ISSN 1078-0947. doi: 10.3934/dcds.2026098. URLhttps://www. aimsciences.org/article/id/69fb06a264170a12e9f7a633. 32

    work page doi:10.3934/dcds.2026098 2026

  6. [6]

    Biagi, D

    S. Biagi, D. Mugnai, and E. Vecchi. A Brezis-Oswald approach for mixed local and nonlocal operators.Commun. Contemp. Math., 26(2):Paper No. 2250057, 28, 2024. ISSN 0219-1997,1793-

    2024

  7. [7]

    URLhttps://doi.org/10.1142/S0219199722500572

    doi: 10.1142/S0219199722500572. URLhttps://doi.org/10.1142/S0219199722500572. 14

    work page doi:10.1142/s0219199722500572

  8. [8]

    Biagi, S

    S. Biagi, S. Dipierro, E. Valdinoci, and E. Vecchi. A Brezis-Nirenberg type result for mixed local and nonlocal operators.NoDEA Nonlinear Differential Equations Appl., 32(4):Paper No. 62, 28,

  9. [9]

    doi: 10.1007/s00030-025-01068-0

    ISSN 1021-9722,1420-9004. doi: 10.1007/s00030-025-01068-0. URLhttps://doi.org/10. 1007/s00030-025-01068-0. 5

    work page doi:10.1007/s00030-025-01068-0

  10. [10]

    A. Biswas. The Pohozaev identity for mixed local-nonlocal operators.J. Math. Anal. Appl., 557 (1):Paper No. 130270, 19, 2026. ISSN 0022-247X,1096-0813. doi: 10.1016/j.jmaa.2025.130270. URLhttps://doi.org/10.1016/j.jmaa.2025.130270. 3

    work page doi:10.1016/j.jmaa.2025.130270 2026

  11. [11]

    Br´ ezis and L

    H. Br´ ezis and L. Nirenberg. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents.Comm. Pure Appl. Math., 36(4):437–477, 1983. ISSN 0010-3640,1097-0312. doi: 10.1002/cpa.3160360405. URLhttps://doi.org/10.1002/cpa.3160360405. 4 36M. Bhakta, N. Biswas, and P. Das

    work page doi:10.1002/cpa.3160360405 1983

  12. [12]

    Byun and K

    S.-S. Byun and K. Song. Mixed local and nonlocal equations with measure data.Calc. Var. Partial Differential Equations, 62(1):Paper No. 14, 35, 2023. ISSN 0944-2669,1432-0835. doi: 10.1007/s00526-022-02349-7. URLhttps://doi.org/10.1007/s00526-022-02349-7. 6, 17

    work page doi:10.1007/s00526-022-02349-7 2023

  13. [13]

    Caffarelli and L

    L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian.Comm. Partial Differential Equations, 32(7-9):1245–1260, 2007. ISSN 0360-5302,1532-4133. doi: 10.1080/ 03605300600987306. URLhttps://doi.org/10.1080/03605300600987306. 8

    work page doi:10.1080/03605300600987306 2007

  14. [14]

    D. Cao, S. Peng, and S. Yan. Infinitely many solutions for p-laplacian equation involving critical sobolev growth.J. Funct. Anal., 262(6):2861–2902, 2012. ISSN 0022-1236,1096-0783. doi: 10. 1016/j.jfa.2012.01.006. URLhttps://doi.org/10.1016/j.jfa.2012.01.006. 5, 6, 7, 27

    work page doi:10.1016/j.jfa.2012.01.006 2012

  15. [15]

    Chakraborty, D

    S. Chakraborty, D. Gupta, S. Malhotra, and K. Sreenadh. Global compactness result for a br´ ezis- nirenberg-type problem involving mixed local nonlocal operator, 2025. URLhttps://arxiv. org/abs/2504.15968. 5, 10, 11

    work page arXiv 2025

  16. [16]

    J. a. V. da Silva, A. Fiscella, and V. A. B. Viloria. Mixed local-nonlocal quasilinear problems with critical nonlinearities.J. Differential Equations, 408:494–536, 2024. ISSN 0022-0396,1090-2732. doi: 10.1016/j.jde.2024.07.028. URLhttps://doi.org/10.1016/j.jde.2024.07.028. 5

    work page doi:10.1016/j.jde.2024.07.028 2024

  17. [17]

    Devillanova and S

    G. Devillanova and S. Solimini. Concentration estimates and multiple solutions to elliptic problems at critical growth.Adv. Differential Equations, 7(10):1257–1280, 2002. ISSN 1079-9389. 4, 5, 7

    2002

  18. [18]

    Domokos and M

    A. Domokos and M. M. Marsh. Projections onto cones in Banach spaces.Fixed Point Theory, 19 (1):167–177, 2018. ISSN 1583-5022,2066-9208. 29

    2018

  19. [19]

    Fortunato and E

    D. Fortunato and E. Jannelli. Infinitely many solutions for some nonlinear elliptic prob- lems in symmetrical domains.Proc. Roy. Soc. Edinburgh Sect. A, 105:205–213, 1987. ISSN 0308-2105,1473-7124. doi: 10.1017/S0308210500022046. URLhttps://doi.org/10.1017/ S0308210500022046. 4

    work page doi:10.1017/s0308210500022046 1987

  20. [20]

    Gao and Y

    F. Gao and Y. Guo. Multiple solutions for quasilinear elliptic equations with critical exponents inR N.Pacific J. Math., 310(1):49–83, 2021. ISSN 0030-8730,1945-5844. doi: 10.2140/pjm.2021. 310.49. URLhttps://doi.org/10.2140/pjm.2021.310.49. 5

    work page doi:10.2140/pjm.2021 2021

  21. [21]

    Garain and J

    P. Garain and J. Kinnunen. On the regularity theory for mixed local and nonlocal quasilinear elliptic equations.Trans. Amer. Math. Soc., 375(8):5393–5423, 2022. ISSN 0002-9947,1088-6850. doi: 10.1090/tran/8621. URLhttps://doi.org/10.1090/tran/8621. 6, 18

    work page doi:10.1090/tran/8621 2022

  22. [22]

    Giaquinta and E

    M. Giaquinta and E. Giusti. On the regularity of the minima of variational integrals.Acta Math., 148:31–46, 1982. ISSN 0001-5962,1871-2509. doi: 10.1007/BF02392725. URLhttps: //doi.org/10.1007/BF02392725. 35

    work page doi:10.1007/bf02392725 1982

  23. [23]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger.Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. ISBN 3-540-41160-7. Reprint of the 1998 edition. 23

    2001

  24. [24]

    T. Gou. Non-degeneracy and uniqueness of ground states to nonlinear elliptic equations with mixed local and nonlocal operators, 2025. URLhttps://arxiv.org/abs/2509.25677. 6

    work page arXiv 2025

  25. [25]

    L. Li, J. Sun, and S. Tersian. Infinitely many sign-changing solutions for the Br´ ezis-Nirenberg problem involving the fractional Laplacian.Fract. Calc. Appl. Anal., 20(5):1146–1164, 2017. ISSN 1311-0454,1314-2224. doi: 10.1515/fca-2017-0061. URLhttps://doi.org/10.1515/ fca-2017-0061. 5

    work page doi:10.1515/fca-2017-0061 2017

  26. [26]

    Ros-Oton and J

    X. Ros-Oton and J. Serra. Nonexistence results for nonlocal equations with critical and su- percritical nonlinearities.Comm. Partial Differential Equations, 40(1):115–133, 2015. ISSN 0360-5302,1532-4133. doi: 10.1080/03605302.2014.918144. URLhttps://doi.org/10.1080/ 03605302.2014.918144

    work page doi:10.1080/03605302.2014.918144 2015

  27. [27]

    Schechter and W

    M. Schechter and W. Zou. On the Br´ ezis-Nirenberg problem.Arch. Ration. Mech. Anal., 197 (1):337–356, 2010. ISSN 0003-9527,1432-0673. doi: 10.1007/s00205-009-0288-8. URLhttps: //doi.org/10.1007/s00205-009-0288-8. 4, 7, 28, 31

    work page doi:10.1007/s00205-009-0288-8 2010

  28. [28]

    Servadei and E

    R. Servadei and E. Valdinoci. A Brezis-Nirenberg result for non-local critical equations in low dimension.Commun. Pure Appl. Anal., 12(6):2445–2464, 2013. ISSN 1534-0392,1553-5258. doi: 10.3934/cpaa.2013.12.2445. URLhttps://doi.org/10.3934/cpaa.2013.12.2445. 5 Mixed local nonlocal Brezis-Nirenberg problem37

    work page doi:10.3934/cpaa.2013.12.2445 2013

  29. [29]

    Servadei and E

    R. Servadei and E. Valdinoci. The Brezis-Nirenberg result for the fractional Laplacian. Trans. Amer. Math. Soc., 367(1):67–102, 2015. ISSN 0002-9947,1088-6850. doi: 10.1090/ S0002-9947-2014-05884-4. URLhttps://doi.org/10.1090/S0002-9947-2014-05884-4. 5

    work page doi:10.1090/s0002-9947-2014-05884-4 2015

  30. [30]

    P. N. Srikanth. Uniqueness of solutions of nonlinear Dirichlet problems.Differential Integral Equations, 6(3):663–670, 1993. ISSN 0893-4983. 4

    1993

  31. [31]

    Struwe.Variational methods, volume 34 ofErgebnisse der Mathematik und ihrer Grenzgebi- ete

    M. Struwe.Variational methods, volume 34 ofErgebnisse der Mathematik und ihrer Grenzgebi- ete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, fourth edition, 2008. ISBN 978-3-540-74012-4. Applications to nonlinear partial differ...

    2008

  32. [32]

    X. Su, E. Valdinoci, Y. Wei, and J. Zhang. On some regularity properties of mixed local and nonlocal elliptic equations.J. Differential Equations, 416:576–613, 2025. ISSN 0022-0396,1090-

    2025

  33. [33]

    URLhttps://doi.org/10.1016/j.jde.2024.10.003

    doi: 10.1016/j.jde.2024.10.003. URLhttps://doi.org/10.1016/j.jde.2024.10.003. 21, 29

    work page doi:10.1016/j.jde.2024.10.003 2024

  34. [34]

    G. Talenti. Best constant in Sobolev inequality.Ann. Mat. Pura Appl. (4), 110:353–372, 1976. ISSN 0003-4622. doi: 10.1007/BF02418013. URLhttps://doi.org/10.1007/BF02418013. 10

    work page doi:10.1007/bf02418013 1976

  35. [35]

    Tan and J

    J. Tan and J. Xiong. A Harnack inequality for fractional Laplace equations with lower order terms.Discrete Contin. Dyn. Syst., 31(3):975–983, 2011. ISSN 1078-0947,1553-5231. doi: 10. 3934/dcds.2011.31.975. URLhttps://doi.org/10.3934/dcds.2011.31.975. 9, 27

    work page doi:10.3934/dcds.2011.31.975 2011

  36. [36]

    S. Yan, J. Yang, and X. Yu. Equations involving fractional Laplacian operator: compactness and application.J. Funct. Anal., 269(1):47–79, 2015. ISSN 0022-1236,1096-0783. doi: 10.1016/j.jfa. 2015.04.012. URLhttps://doi.org/10.1016/j.jfa.2015.04.012. 5, 6, 7 Department of Mathematics, Indian Institute of Science Education and Research (IISER-Pune), Dr. Homi...

    work page doi:10.1016/j.jfa 2015