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Multiplicity of positive solutions for the fractional Schrödinger–Poisson system with critical nonlocal term

Xilin Dou

https://orcid.org/0009-0008-7619-9999

College of Science, Minzu University of China, Beijing 100081, P. R. China

E-mail Address: xldou816@163.com

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Xiaoming He

https://orcid.org/0000-0001-5846-4009

College of Science, Minzu University of China, Beijing 100081, P. R. China

E-mail Address: xmhe923@muc.edu.cn

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, and

Vicenţiu D. Rădulescu

https://orcid.org/0000-0003-4615-5537

Faculty of Applied Mathematics, AGH University of Kraków, al. Mickiewicza 30, 30-059 Kraków, Poland

Faculty of Electrical Engineering and Communication, Brno University of Technology Technická 3058/10, Brno 61600, Czech Republic

Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania

Simion Stoilow Institute of Mathematics of the Romanian Academy, Calea Griviţei No. 21, 010702, Bucharest, Romania

School of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China

E-mail Address: radulescu@inf.ucv.ro

Corresponding author.

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https://doi.org/10.1142/S1664360723500121Cited by:16 (Source: Crossref)

Abstract

This paper deals with the following fractional Schrödinger–Poisson system:

with multiple competing potentials and a critical nonlocal term, where

$s \in (0, 1)$ ,

$q \in (1, 2)$ or

$q \in (4, 2_{s}^{*})$ , and

$2_{s}^{*} = \frac{6}{3 - 2 s}$ is the fractional critical exponent. By combining the Nehari manifold analysis and the Ljusternik–Schnirelmann category theory, we establish how the coefficient

$K$ of the nonlocal critical nonlinearity affects the number of positive solutions. We propose a new relation between the number of positive solutions and the category of the global maximal set of

$K$ .

Communicated by Neil Trudinger

Keywords:

AMSC: 35J62, 35J50, 35B65

References

1. A. Ambrosetti , On Schrödinger-Poisson systems, Milan J. Math. 76 (2008) 257–274. Crossref, Google Scholar
2. V. Ambrosio and T. Isernia , Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional $p$ -Laplacian, Discrete Contin. Dyn. Syst. 38 (2018) 5835–5881. Crossref, Google Scholar
3. V. Ambrosio, T. Isernia and V. D. Rădulescu , Concentration of positive solutions for a class of fractional p-Kirchhoff type equations, Proc. Roy. Soc. Edinburgh Sect. A 151(2) (2021) 601–651. Crossref, Google Scholar
4. C. O. Alves, M. A. S. Souto and S. H. Soares , Schrödinger–Poisson equations without Ambrosetti–Rabinowitz condition, J. Math. Anal. Appl. 377 (2011) 584–592. Crossref, Google Scholar
5. A. Azzollini, P. d’Avenia and G. Vaira , Generalized Schrödinger–Newton system in dimension $N \geq 3$ : Critical case, J. Math. Anal. Appl. 449 (2017) 531–552. Crossref, Google Scholar
6. V. Benci and G. Cerami , Existence of positive solutions of the equation $- Δ u + a (x) u = u^{(N + 2) / (N - 2)}$ in $ℝ^{N}$ , J. Funct. Anal. 88 (1990) 90–117. Crossref, Google Scholar
7. V. Benci and D. Fortunato , An eigenvalue problem for the Schrödinger–Maxwell equation, Topol. Methods Nonlinear Anal. 11 (1998) 283–293. Crossref, Google Scholar
8. V. Benci and D. Fortunato , Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations, Notices Amer. Math. Soc. 51 (2004) 1336–1347. Google Scholar
9. L. A. Caffarelli and L. Silvestre , An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007) 1245–1260. Crossref, Google Scholar
10. G. Cerami and G. Vaira , Positive solutions for some non-autonomous Schrödinger–Poisson systems, J. Differential Equations 248 (2010) 521–543. Crossref, Google Scholar
11. S. Cingolani and M. Lazzo , Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations 160 (2000) 118–138. Crossref, Google Scholar
12. T. D’Aprile and D. Mugnai , Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 893–906. Crossref, Google Scholar
13. J. M. do Ó, O. H. Miyagaki and M. Squassina , Critical and subcritical fractional problems with vanishing potentials, Commun. Contemp. Math. 18 (2016) 1550063. Link, Google Scholar
14. X. Dou and X. He , Ground states for critical fractional Schrödinger–Poisson systems with vanishing potentials, Math. Methods Appl. Sci. 45 (2022) 9089–9110. Crossref, Google Scholar
15. H. Fan, Z. Feng and X. Yan , Positive solutions of fractional Schrödinger–Poisson systems involving critical nonlinearities with potential, Differential Integral Equations 35 (2022) 1–56. Crossref, Google Scholar
16. X. Feng , Existence and concentration of ground state solutions for doubly critical Schrödinger–Poisson-type systems, Z. Angew. Math. Phys. 71(5) (2020), Paper No. 154, 25 pp. Crossref, Google Scholar
17. X. Feng , Nontrivial solution for Schrödinger–Poisson equations involving the fractional Laplacian with critical exponent, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 115(1) (2021), Paper No. 10, 19 pp. Google Scholar
18. D. Goel and K. Sreenadh , Kirchhoff equations with Hardy–Littlewood–Sobolev critical nonlinearity, Nonlinear Anal. 186 (2019) 162–186. Crossref, Google Scholar
19. X. He , Positive solutions for fractional Schrödinger–Poisson systems with doubly critical exponents, Appl. Math. Lett. 120 (2021) 107190. Crossref, Google Scholar
20. X. He and W. Zou , Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations 55 (2016) 1–39. Crossref, Google Scholar
21. N. Laskin , Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000) 298–305. Crossref, Google Scholar
22. N. Laskin , Fractional Schrödinger equation, Phys. Rev. E. 66 (2002) 56–108. Crossref, Google Scholar
23. C. Lei, V. D. Rădulescu and B. Zhang , Groundstates of the Schrödinger–Poisson–Slater equation with critical growth, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 117(3) (2023), Paper No. 128, 34 pp. Google Scholar
24. K. Li, Bound state solutions for non-autonomous fractional Schrödinger–Poisson equations with critical exponent, preprint (2019), arXiv:1905.11643 [math.AP]. Google Scholar
25. N. Li and X. He , Existence and multiplicity results for some Schrödinger–Poisson system with critical growth, J. Math. Anal. Appl. 488 (2020) 124071. Crossref, Google Scholar
26. F. Li, Y. Li and J. Shi , Existence of positive solutions to Schrödinger–Poisson type systems with critical exponent, Commun. Contemp. Math. 16 (2014) 1450036. Link, Google Scholar
27. F. Li, Y. Li and J. Shi , Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term, Calc. Var. Partial Differential Equations 56 (2017) 134. Crossref, Google Scholar
28. Y. Li, B. Zhang and X. Han , Existence and concentration behavior of positive solutions to Schrödinger–Poisson–Slater equations, Adv. Nonlinear Anal. 12(1) (2023), Paper No. 20220293, 23 pp. Google Scholar
29. E. Lieb and M. Loss , Analysis, Graduate Studies in Mathematics, Vol. 14 (American Mathematical Society, Providence, RI, 2001). Crossref, Google Scholar
30. P. L. Lions , The concentration-compactness principle in the calculus of variation. The locally compact case. Part I, Ann. Inst. H. Poincaré, Anal. Non Linéaire 1 (1984) 223–283. Crossref, Google Scholar
31. P. L. Lions , The concentration-compactness principle in the calculus of variation. The locally compact case. Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 109–145. Crossref, Google Scholar
32. H. Liu , Positive solutions of an asymptotically periodic Schrödinger–Poisson system with critical exponent, Nonlinear Anal. Real World Appl. 32 (2016) 198–212. Crossref, Google Scholar
33. Z. Liu and J. Zhang , Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth, ESAIM: Control Optim. Calc. Var. 23 (2017) 1515–1542. Crossref, Google Scholar
34. H. Luo and X. Tang , Ground state and multiple solutions for the fractional Schrödinger–Poisson system with critical Sobolev exponent, Nonlinear Anal. Real World Appl. 42 (2018) 24–52. Crossref, Google Scholar
35. G. Molica Bisci, V. D. Rădulescu and R. Servadei , Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, Vol. 162 (Cambridge University Press, Cambridge, 2016). Crossref, Google Scholar
36. D. Mugnai , Non-existence results for the coupled Klein–Gordon–Maxwell equations, Adv. Nonlinear Stud. 4 (2004) 307–322. Crossref, Google Scholar
37. E. Murcia and G. Siciliano , Positive semiclassical states for a fractional Schrödinger–Poisson system, Differential Integral Equations 30 (2017) 231–258. Crossref, Google Scholar
38. S. Qu and X. He , On the number of concentrating solutions of a fractional Schrödinger–Poisson system with doubly critical growth, Anal. Math. Phys. 12(2) (2022), Paper No. 59, 49 pp. Crossref, Google Scholar
39. D. Ruiz , The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006) 655–674. Crossref, Google Scholar
40. S. Secchi , Ground state solutions for nonlinear fractional Schrödinger equations in $ℝ^{N}$ , J. Math. Phys. 54 (2013) 031501, https://doi.org/10.1063/1.4793990. Crossref, Google Scholar
41. R. Servadei and E. Valdinoci , The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015) 67–102. Crossref, Google Scholar
42. K. Teng , Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent, J. Differential Equations 261 (2016) 3061–3106. Crossref, Google Scholar
43. M. Willem , Minimax Theorems (Birkhäuser, Boston, 1996). Crossref, Google Scholar
44. M. Willem , Functional Analysis, Fundamentals and Applications, Cornerstones (Birkhäuser/Springer, New York, 2013). Crossref, Google Scholar
45. Z. Yang, Y. Yu and F. Zhao , Concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system involving critical exponent, Commun. Contemp. Math. 21(6) (2019) 1850027. Link, Google Scholar
46. R. Yin, J. Zhang and X. Shang , Positive ground state solutions for Schrödinger–Poisson system with critical nonlocal term in $ℝ^{3}$ , Math. Methods Appl. Sci. 43 (2020) 8736–8752. Crossref, Google Scholar
47. Y. Yu, F. Zhao and L. Zhao , The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system, Calc. Var. Partial Differential Equations 56 (2017), Art. 116, 25 pp. Crossref, Google Scholar
48. J. Zhang, J. M. do Ó and M. Squassina , Fractional Schrödinger–Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud. 16 (2016) 15–30. Crossref, Google Scholar
49. W. Zhang, J. Zhang and V. D. Rădulescu , Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction, J. Differential Equations 347 (2023) 56–103. Crossref, Google Scholar
50. L. Zhao and F. Zhao , Positive solutions for Schrödinger–Poisson equations with a critical exponent, Nonlinear Anal. 70 (2009) 2150–2164. Crossref, Google Scholar

Vol. 14, No. 02

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History

Received 10 March 2023

Revised 31 July 2023

Accepted 10 August 2023

Published: 12 September 2023

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This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Multiplicity of positive solutions for the fractional Schrödinger–Poisson system with critical nonlocal term

Abstract

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