Quasilinear elliptic equations with unbalanced growth and singular perturbation

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, we study parametric quasilinear elliptic equations driven by the double phase operator, where the right-hand side consists of a singular term and a sublinear term. By combining a new Hopf’s Lemma with truncation techniques and an abstract critical point theorem, we establish the existence of three bounded positive solutions and provide an explicit upper bound for the parameter.

Keywords

double phase operator / Hopf’s Lemma / multiple solutions / singular problems / unbalanced growth

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv., 29 (1987), 675-710. https://doi.org/10.1070/IM1987v029n01ABEH000958

[2] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differ. Equations, 90 (1991), 1-30. https://doi.org/10.1016/0022-0396(91)90158-6

[3] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284. https://doi.org/10.1007/BF00251503

[4] W. Liu, G. Dai, Three ground state solutions for double phase problem, J. Math. Phys., 59 (2018), 121503. https://doi.org/10.1063/1.5055300

[5] P. Baroni, M. Colombo, G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. Theory Methods Appl., 121 (2015), 206-222. https://doi.org/10.1016/j.na.2014.11.001

[6] P. Baroni, M. Colombo, G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J., 27 (2016), 347-379. https://doi.org/10.1090/spmj/1392

[7] P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differ. Equations, 57 (2018), 62. https://doi.org/10.1007/s00526-018-1332-z

[8] P. Baroni, T. Kuusi, G. Mingione, Borderline gradient continuity of minima, J. Fixed Point Theory Appl., 15 (2014), 537-575. https://doi.org/10.1007/s11784-014-0188-x

[9] S. S. Byun, J. Oh, Regularity results for generalized double phase functionals, Anal. PDE, 13 (2020), 1269-1300. https://doi.org/10.2140/apde.2020.13.1269

[10] M. Colombo, G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273. https://doi.org/10.1007/s00205-015-0859-9

[11] M. Colombo, G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496. https://doi.org/10.1007/s00205-014-0785-2

[12] C. De Filippis, G. Palatucci, Hölder regularity for nonlocal double phase equations, J. Differ. Equations, 267 (2019), 547-586. https://doi.org/10.1016/j.jde.2019.01.017

[13] J. Ok, Regularity for double phase problems under additional integrability assumptions, Nonlinear Anal., 194 (2020), 111408. https://doi.org/10.1016/j.na.2018.12.019

[14] M. A. Ragusa, A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal., 9 (2020), 710-728. https://doi.org/10.1515/anona-2020-0022

[15] J. A. D. Ackroyd, On the laminar compressible boundary layer with stationary origin on a moving flat wall, Math. Proc. Cambridge Philos. Soc., 63 (1967), 871-888. https://doi.org/10.1017/S0305004100041840

[16] R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts , Clarendon Press, Oxford University Press, 1975.

[17] M. G. Crandall, P. H. Rabinowitz, L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Partial Differ. Equations, 2 (1977), 193-222. https://doi.org/10.1080/03605307708820029

[18] J. Shi, M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. Edinburgh Math. Soc. Sect. A: Math., 128 (1998), 1389-1401. https://doi.org/10.1017/S0308210500027384

[19] Y. Sun, S. Wu, Y. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differ. Equations, 176 (2001), 511-531. https://doi.org/10.1006/jdeq.2000.3973

[20] Y. Sun, S. Wu, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257-1284. https://doi.org/10.1016/j.jfa.2010.11.018

[21] W. Liu, G. Dai, N. S. Papageorgiou, P. Winkert, Existence of solutions for singular double phase problems via the Nehari manifold method, Anal. Math. Phys., 12 (2022), 75. https://doi.org/10.1007/s13324-022-00686-6

[22] Y. Bai, L. Gasiński, N. S. Papageorgiou, Singular double phase equations with a sign changing reaction, Commun. Nonlinear Sci. Numer. Simul., 142 (2025), 108566. https://doi.org/10.1016/j.cnsns.2024.108566

[23] Z. Liu, N. S. Papageorgiou, Singular double phase equations, Acta Math. Sci., 43 (2023), 1031-1044. https://doi.org/10.1007/s10473-023-0304-3

[24] N. S. Papageorgiou, D. D. Repovš, C. Vetro, Positive solutions for singular double phase problems, J. Math. Anal. Appl., 501 (2021), 123896. https://doi.org/10.1016/j.jmaa.2020.123896

[25] W. Liu, P. Winkert, Combined effects of singular and superlinear nonlinearities in singular double phase problems in R^N , J. Math. Anal. Appl., 507 (2022), 125762. https://doi.org/10.1016/j.jmaa.2021.125762

[26] Y. Bai, N. S. Papageorgiou, S. Zeng, Parametric singular double phase Dirichlet problems, Adv. Nonlinear Anal., 12 (2023), 20230122. https://doi.org/10.1515/anona-2023-0122

[27] N. S. Papageorgiou, V. D. Rădulescu, Y. Zhang, Anisotropic singular double phase Dirichlet problems, Discrete Contin. Dyn. Syst. - Ser. S, 14 (2021), 4465-4502. https://doi.org/10.3934/dcdss.2021111

[28] G. Failla, L. Gasiński, N. S. Papageorgiou, M. Skupień, Existence and uniqueness of solutions for singular double phase equations, Appl. Anal., (2025), 1-16. https://doi.org/10.1080/00036811.2025.2515607

[29] N. S. Papageorgiou, V. D. Rădulescu, S. Yuan, Nonautonomous double-phase equations with strong singularity and concave perturbation, Bull. London Math. Soc., 56 (2024), 1245-1262. https://doi.org/10.1112/blms.12991

[30] N. S. Papageorgiou, V. D. Rădulescu, Y. Zhang, Strongly singular double phase problems, Mediterr. J. Math., 19 (2022), 82. https://doi.org/10.1007/s00009-022-02013-6

[31] A. Arora, G. Dwivedi, Existence results for singular double phase problem with variable exponents, Mediterr. J. Math., 20 (2023), 155. https://doi.org/10.1007/s00009-023-02366-6

[32] R. Arora, A. Fiscella, T. Mukherjee, P. Winkert, On double phase Kirchhoff problems with singular nonlinearity, Adv. Nonlinear Anal., 12 (2023), 20220312. https://doi.org/10.1515/anona-2022-0312

[33] A. Bahrouni, V.D. Rădulescu, D. D. Repovš, Nonvariational and singular double phase problems for the Baouendi-Grushin operator, J. Differ. Equations, 303 (2021), 645-666. https://doi.org/10.1016/j.jde.2021.09.033

[34] C. Farkas, P. Winkert, An existence result for singular Finsler double phase problems, J. Differ. Equations, 286 (2021), 455-473. https://doi.org/10.1016/j.jde.2021.03.036

[35] P. Garain, T. Mukherjee, On an anisotropic double phase problem with singular and sign changing nonlinearity, Nonlinear Anal. Real World Appl., 70 (2023), 103790. https://doi.org/10.1016/j.nonrwa.2022.103790

[36] U. Guarnotta, P. Winkert, Degenerate singular Kirchhoff problems in Musielak-Orlicz spaces, Nonlinear Anal., 264 (2026), 113986. https://doi.org/10.1016/j.na.2025.113986

[37] J. Han, Z. Liu, N. S. Papageorgiou, A singular double phase eigenvalue problem with a superlinear indefinite perturbation, Nonlinear Anal., 262 (2026), 113941. https://doi.org/10.1016/j.na.2025.113941

[38] N. S. Papageorgiou, Z. Peng, Singular double phase problems with convection, Nonlinear Anal. Real World Appl., 81 (2025), 104213. https://doi.org/10.1016/j.nonrwa.2024.104213

[39] M. T. O. Pimenta, P. Winkert, Strongly singular problems with unbalanced growth, Ann. Mat. Pura Appl., 204 (2025), 2129-2145. https://doi.org/10.1007/s10231-025-01564-1

[40] I. Sim, B. Son, On a class of singular double phase problems with nonnegative weights whose sum can be zero, Nonlinear Anal., 237 (2023), 113384. https://doi.org/10.1016/j.na.2023.113384

[41] L. Zhao, Y. He, P. Zhao, The existence of three positive solutions of a singular p-Laplacian problem, Nonlinear Anal. Theory Methods Appl., 74 (2011), 5745-5753. https://doi.org/10.1016/j.na.2011.05.065

[42] G. Bonanno, G. M. Bisci, Three weak solutions for elliptic Dirichlet problems, J. Math. Anal. Appl., 382 (2011), 1-8. https://doi.org/10.1016/j.jmaa.2011.04.026

[43] A. Kristály, H. Lisei, C. Varga, Multiple solutions for p-Laplacian type equations, Nonlinear Anal. Theory Methods Appl., 68 (2008), 1375-1381. https://doi.org/10.1016/j.na.2006.12.031

[44] G. Bonanno, G. M. Bisci, V. D. Rădulescu, Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces, Nonlinear Anal. Theory Methods Appl., 74 (2011), 4785-4795. https://doi.org/10.1016/j.na.2011.04.049

[45] Á. Crespo-Blanco, L. Gasiński, P. Harjulehto, P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differ. Equations, 323 (2022), 182-228. https://doi.org/10.1016/j.jde.2022.03.029

[46] W. Liu, G. Dai, Existence and multiplicity results for double phase problem, J. Differ. Equations, 265 (2018), 4311-4334. https://doi.org/10.1016/j.jde.2018.06.006

[47] N. S. Papageorgiou, P. Winkert, Applied Nonlinear Functional Analysis , 2nd edition, De Gruyter, Berlin, 2024. https://doi.org/10.1515/9783111286952

[48] G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal. Theory Methods Appl., 54 (2003), 651-665. https://doi.org/10.1016/S0362-546X(03)00092-0

[49] B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. Theory Methods Appl., 74 (2011), 7446-7454. https://doi.org/10.1016/j.na.2011.07.064

[50] B. Ricceri, On a three critical points theorem, Arch. Math., 75 (2000), 220-226. https://doi.org/10.1007/s000130050496