Abstract
In this paper we consider the following fractional Schrödinger–Poisson system
where \(s\in (\frac{1}{2},1)\) and g(x, u) is asymptotically linear at infinity. Under certain assumptions on K(x) and g(x, u), we prove the existence of ground state solutions by variational methods.
Similar content being viewed by others
References
Ambrosetti, A.: On Schrödinger–Poisson systems. Milan J. Math. 76, 257–274 (2008)
Azzollini, A., Pomponio, A.: Ground state solutions for the nonlinear Schrödinger–Maxwell equations. J. Math. Anal. Appl. 345, 90–108 (2008)
Ambrosetti, A., Ruiz, D.: Mutiple bound states for Schrödinger–Poisson problems. Commun. Contemp. Math. 10, 391–404 (2008)
Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger–Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283–293 (1998)
Benci, V., Fortunato, D.: Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 23–53 (2014)
Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. 367, 911–941 (2015)
Cerami, G., Molle, R.: Positive bound state solutions for some Schrödinger–Poisson systems. Nonlinearity 29, 3103–3119 (2016)
Cerami, G., Vaira, G.: Positive solutions for some non-autonomous Schrödinger–Poisson systems. J. Differ. Equ. 248, 521–543 (2010)
Chang, X., Wang, Z.Q.: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26, 479–494 (2013)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Dávila, J., del Pino, M., Wei, J.: Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Differ. Equ. 256, 858–892 (2014)
Ekeland, I.: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin (1990)
Frank, R., Lenzmann, E.: Uniqueness of nonlinear ground states for fractional Laplacian in \({\mathbb{R}}\). Acta Math. 210, 261–318 (2013)
Frank, R., Lenzmann, E., Silvestre, L.: Uniqueness of radial solutions for the fractional Laplacian. Comm. Pure Appl. Math. 69(9), 1671–1726 (2016)
Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 142, 1237–1262 (2012)
Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)
Liu, W.: Existence of multi-bump solutions for the fractional Schrödinger–Poisson system. J. Math. Phys. 57, 091502 (2016)
Ruiz, D.: The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)
Sun, J., Chen, H., Nieto, J.J.: On ground state solutions for some non-autonomous Schrödinger–Poisson systems. J. Differ. Equ. 252, 3365–3380 (2012)
Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in \({\mathbb{R}}^{N}\). J. Math. Phys. 54, 031501 (2013)
Shang, X., Zhang, J.: Ground states for fractional Schrödinger equations with critical growth. Nonlinearity 27, 187–207 (2014)
Teng, K.: Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent. J. Differ. Equ. 261, 3061–3106 (2016)
Vaira, G.: Ground states for Schrödinger–Poisson type systems. Ric. Mater. 60, 263–297 (2011)
Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
Yu, Y., Zhao, F., Zhao, L.: The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system. Calc. Var. Partial Differ. Equ. 56, 1–25 (2017)
Zhu, H.: Asymptotically linear Schrödinger–Poisson systems with potentials vanishing at infinity. J. Math. Anal. Appl. 380, 501–510 (2011)
Zhang, J., Squassina, M.: Fractional Schrödinger–Poisson systems with a general subcritical or critical nonlinearity. Adv. Nonlinear Stud. 16, 15–30 (2016)
Zhang, J., Liu, X., Jiao, H.: Multiplicity of positive solutions for fractional Laplacian equations involving critical nonlinearity. Topol. Methods Nonlinear Anal. 53, 151–182 (2019)
Acknowledgements
The authors wish to thank the anonymous referees very much for carefully reading this paper and suggesting many valuable comments. P. Chen was supported by the Research Foundation of Education Bureau of Hubei Province, China (Grant No. Q20192505). X. Liu was partially supported by the National Natural Science Foundation of China (Grant No. 11771342).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chen, P., Liu, X. Ground states for asymptotically linear fractional Schrödinger–Poisson systems. J. Pseudo-Differ. Oper. Appl. 12, 8 (2021). https://doi.org/10.1007/s11868-021-00390-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11868-021-00390-2