Existence for critical Schrödinger–Bopp–Podolsky system with p-Laplacian in R³

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Abstract

We intend to study a class of the critical Schrödinger–Bopp–Podolsky system with p-Laplacian in R³. Under different perturbation terms, the existence and multiplicity of nontrivial solutions are obtained by using critical point theorem. Considering the influence of the p-Laplacian operator and critical and nonlocal terms, which cause the loss of the compactness condition, we attempt to address this difficulty by using the concentration-compactness principle and some clever analysis.

Keywords

Schrödinger–Bopp–Podolsky system / critical growth / p-Laplacian / choquard nonlinearity / variational methods / concentration-compactness principles

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References

[1] P. d'Avenia, G. Siciliano, Nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics: Solutions in the electrostatic case, J. Differ. Equations, 267 (2019), 1025-1065. https://doi.org/10.1016/j.jde.2019.02.001

[2] F. Bopp, Eine lineare theorie des elektrons, Ann. Phys., 430 (1940), 345-384. https://doi.org/10.1002/andp.19404300504

[3] B. Podolsky, A generalized electrodynamics part I-Non-Quantum, Phys. Rev., 62 (1942), 68-71. https://doi.org/10.1103/PhysRev.62.68

[4] J. Frenkel, 4/3 problem in classical electrodynamics, Phys. Rev. E., 54 (1996), 5859-5862. https://doi.org/10.1103/PhysRevE.54.5859

[5] M. C. Bertin, B. M. Pimentel, C. E. Valccel, G. E. R. Zambrano, Hamilton-Jacobi formalism for Podolsky's electromagnetic theory on the null-plane, J. Math. Phys., 58 (2017), 082902. https://doi.org/10.1063/1.4999846

[6] R. Bufalo, B. M. Pimentel, D. E. Soto, Causal approach for the electron-positron scattering in generalized quantum electrodynamics, Phys. Rev. D., 90 (2014), 085012. https://doi.org/10.1103/PhysRevD.90.085012

[7] R. Bufalo, B. M. Pimentel, D. E. Soto, Normalizability analysis of the generalized quantum electrodynamics from the causal point of view, Int. J. Mod. Phys. A, 32 (2017), 1750165. https://doi.org/10.1142/S0217751X17501652

[8] G. Siciliano, K. Silva, The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field, Publ. Math., 64 (2020), 373-390. https://doi.org/10.5565/PUBLMAT6422001

[9] H. M. Damian, G. Siciliano, Critical Schrödinger-Bopp-Podolsky systems: solutions in the semiclassical limit, Calc. Var. Partial Differ. Equations, 63 (2024), 155. https://doi.org/10.1007/s00526-024-02775-9

[10] S. Liu, H. Chen, Existence and asymptotic behaviour of positive ground state solution for critical Schrödinger-Bopp-Podolsky system, Electron. Res. Arch., 30 (2022), 2138-2164. https://doi.org/10.3934/era.2022108

[11] Y. Li, X. Chang, Z. Feng, Normalized solutions for Sobolev critical Schrödinger-Bopp-Podolsky systems, Electron. J. Differ. Equations, 2023 (2023), 1-19. https://doi.org/10.58997/ejde.2023.56

[12] H. R. Quoirin, G. Siciliano, K. Silva, Critical points at prescribed energy level for Schrödinger-Bopp-Podolsky systems, Mat. Contemp., 56 (2023), 4-19. http://doi.org/10.21711/231766362023/rmc562

[13] Z. Shen, N. Yang, Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity, Adv. Nonlinear Anal., 13 (2024), 20240008. https://doi.org/10.1515/anona-2024-0008

[14] Y. Xiao, S. Chen, M. Shu, The existence of ground state solutions for a Schrödinger-Bopp-Podolsky system with convolution nonlinearity, J. Geom. Anal., 33 (2023), 374. https://doi.org/10.1007/s12220-023-01437-0

[15] R. L. Alves, Generalized Schrödinger-Bopp-Podolsky type system with singular nonlinearity, J. Math. Anal. Appl., 539 (2024), 128499. https://doi.org/10.1016/j.jmaa.2024.128499

[16] Q. Li, V. D. Rădulescu, W. Zhang, Normalized ground states for the Sobolev critical Schrödinger equation with at least mass critical growth, Nonlinearity, 37 (2024), 025018. https://doi.org/10.1088/1361-6544/ad1b8b

[17] S. Liang, J. Ma, S. Shi, Y. Song, Multiple normalized solutions for Choquard equation involving the biharmonic operator and competing potentials in R^N , Bull. Math. Sci., 15 (2025), 2450017. https://doi.org/10.1142/s1664360724500176

[18] R. Zhang, S. Yao, J. Sun, Normalized solutions for Schrödinger-Bopp-Podolsky system with a negative potential, Appl. Math. Lett., 161 (2025), 109368. https://doi.org/10.1016/j.aml.2024.109368

[19] Z. Zhang, Z. Zhang, Normalized solutions to p-Laplacian equations with combined nonlinearities , Nonlinearity, 35 (2022), 5621-5663. https://doi.org/10.1088/1361-6544/ac902c

[20] Y. Du, J. Su, C. Wang, The Schrödinger-Poisson system with p-Laplacian , Appl. Math. Lett., 120 (2021), 107286. https://doi.org/10.1016/j.aml.2021.107286

[21] Y. Du, J. Su, C. Wang, On a quasilinear Schrödinger-Poisson system, J. Math. Anal. Appl., 505 (2022), 125446. https://doi.org/10.1016/j.jmaa.2021.125446

[22] Y. Du, J. Su, C. Wang, On the critical Schrödinger-Poisson system with p-Laplacian , Commun. Pure Appl. Anal., 21 (2022), 1329-1342. https://doi.org/10.3934/cpaa.2022020

[23] F. Wang, M. Xiang, Multiplicity of solutions for a class of fractional Choquard-Kirchhoff equations involving critical nonlinearity, Anal. Math. Phys., 9 (2019), 1-16. https://doi.org/10.1007/s13324-017-0174-8

[24] M. Xiang, B. Zhang, X. Zhang, A nonhomogeneous fractional p-Kirchhoff type problem involving critical exponent in R^N, Adv. Nonlinear Stud., 17 (2017), 611-640. https://doi/10.1515/ans-2016-6002

[25] K. Perera, Abstract multiplicity theorems and applications to critical growth problems, J. Anal. Math., 157 (2025), 211-223. https://doi.org/10.1007/s11854-025-0389-9

[26] G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. https://doi.org/10.1007/BF02418013

[27] E. H. Lieb, M. Loss, Analysis , 2nd edition, Graduate Studies in Mathematics, 1997. https://doi.org/10.1090/gsm/014

[28] 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), 109245. https://doi.org/10.1016/j.aml.2024.109245

[29] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations , 1984.

[30] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part I and II, Ann. Inst. Henri Poincare C, 1 (1984), 109-145. https://doi.org/10.1016/S0294-1449(16)30428-0

[31] S. Liang, J. Zhang, Multi-bump solutions for a class of Kirchhoff type problems with critical growth in R^N , Topol. Methods Nonlinear Anal., 48 (2016), 71-101. https://doi.org/10.12775/TMNA.2016.040

[32] J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differ. Equations, 3 (1995), 493-512. https://doi.org/10.1007/BF01187898

[33] S. Liang, J. Zhang, On some p-Laplacian equation with electromagnetic fields and critical nonlinearity in R^N , J. Math. Phys., 56 (2015), 041504. https://doi.org/10.1063/1.4918791

[34] E. R. Fadell, P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174. https://doi.org/10.1007/BF01390270