Ground state solutions for the fractional Schrödinger-Poisson system involving doubly critical exponents

Yang Pu; Hongying Li; Jiafeng Liao; Yang Pu; Hongying Li; Jiafeng Liao

doi:10.3934/math.20221008

AIMS Mathematics

2022, Volume 7, Issue 10: 18311-18322. doi: 10.3934/math.20221008

Previous Article Next Article

Research article

Ground state solutions for the fractional Schrödinger-Poisson system involving doubly critical exponents

1.
School of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637009, China
2.
College of Mathematics Education, China West Normal University, Nanchong, Sichuan 637009, China

Received: 20 June 2022 Revised: 01 August 2022 Accepted: 03 August 2022 Published: 15 August 2022
MSC : 35J20, 35J60

In this article, we are dedicated to studying the fractional Schrödinger-Poisson system involving doubly critical exponent. By using the variational method and analytic techniques, we establish the existence of positive ground state solution.
- fractional Schrödinger-Poisson system,
- doubly critical exponent,
- variational method,
- positive solution
Citation: Yang Pu, Hongying Li, Jiafeng Liao. Ground state solutions for the fractional Schrödinger-Poisson system involving doubly critical exponents[J]. AIMS Mathematics, 2022, 7(10): 18311-18322. doi: 10.3934/math.20221008

Related Papers:

Abstract

In this article, we are dedicated to studying the fractional Schrödinger-Poisson system involving doubly critical exponent. By using the variational method and analytic techniques, we establish the existence of positive ground state solution.

References

[1]	N. Landkof, Foundations of modern potential theory, Berlin: Springer-Verlag, 1972.
[2]	L. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425–461. http://dx.doi.org/10.1007/s00222-007-0086-6 doi: 10.1007/s00222-007-0086-6
[3]	L. Caffarelli, J. Roquejoffre, Y. Sire, Variational problems in free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151–1179. http://dx.doi.org/10.4171/JEMS/226 doi: 10.4171/JEMS/226
[4]	N. Laskin, Fractional quantum mechanics and L$\acute{\mathrm{e}}$vy path integrals, Phys. Lett. A, 268 (2000), 298–305. http://dx.doi.org/10.1016/S0375-9601(00)00201-2 doi: 10.1016/S0375-9601(00)00201-2
[5]	L. Caffarelli, E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var., 41 (2011), 203–240. http://dx.doi.org/10.1007/s00526-010-0359-6 doi: 10.1007/s00526-010-0359-6
[6]	H. Liu, Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal.-Real, 32 (2016), 198–212. http://dx.doi.org/10.1016/j.nonrwa.2016.04.007 doi: 10.1016/j.nonrwa.2016.04.007
[7]	J. Liu, C. Ji, Concentration results for a magnetic Schrödinger-Poisson system with critical growth, Adv. Nonlinear Anal., 10 (2021), 775–798. http://dx.doi.org/10.1515/anona-2020-0159 doi: 10.1515/anona-2020-0159
[8]	F. Li, Y. Li, J. Shi, Existence and multiplicity of positive solutions to Schrödinger-Poisson type systems with critical nonlocal term, Calc. Var., 56 (2017), 134. http://dx.doi.org/10.1007/s00526-017-1229-2 doi: 10.1007/s00526-017-1229-2
[9]	X. Feng, Ground state solutions for a class of Schrödinger-Poisson systems with partial potential, Z. Angew. Math. Phys., 71 (2020), 37. http://dx.doi.org/10.1007/s00033-020-1254-4 doi: 10.1007/s00033-020-1254-4
[10]	C. Ji, Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger-Poisson system in $\mathbb{R}^3$, Ann. Mat. Pur. Appl., 198 (2019), 1563–1579. http://dx.doi.org/10.1007/s10231-019-00831-2 doi: 10.1007/s10231-019-00831-2
[11]	K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differ. Equations, 261 (2016), 3061–3106. http://dx.doi.org/10.1016/j.jde.2016.05.022 doi: 10.1016/j.jde.2016.05.022
[12]	K. Teng, R. Agarwal, Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth, Math. Method. Appl. Sci., 41 (2018), 8258–8293. http://dx.doi.org/10.1002/mma.5289 doi: 10.1002/mma.5289
[13]	Y. Yu, F. Zhao, L. Zhao, The concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system, Calc. Var., 56 (2017), 116. http://dx.doi.org/10.1007/s00526-017-1199-4 doi: 10.1007/s00526-017-1199-4
[14]	W. Long, J. Yang, W. Yu, Nodal solutions for fractional Schrödinger-Poisson problems, Sci. China Math., 63 (2020), 2267–2286. http://dx.doi.org/10.1007/s11425-018-9452-y doi: 10.1007/s11425-018-9452-y
[15]	X. Feng, X. Yang, Existence of ground state solutions for fractional Schrödinger-Poisson systems with doubly critical Growth, Mediterr. J. Math., 18 (2021), 41. http://dx.doi.org/10.1007/s00009-020-01660-x doi: 10.1007/s00009-020-01660-x
[16]	X. He, Positive solutions for fractional Schrödinger-Poisson systems with doubly critical exponents, Appl. Math. Lett., 120 (2021), 107190. http://dx.doi.org/10.1016/j.aml.2021.107190 doi: 10.1016/j.aml.2021.107190
[17]	H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490.
[18]	X. Feng, Ground state solutions for Schrödinger-Poisson systems involving the fractional Laplacian with critical exponent, J. Math. Phys., 60 (2019), 051511. http://dx.doi.org/10.1063/1.5088710 doi: 10.1063/1.5088710
[19]	R. Servafei, E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, T. Am. Math. Soc., 367 (2015), 67–102.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)