# American Institute of Mathematical Sciences

February  2022, 5(1): 57-66. doi: 10.3934/mfc.2021023

## Bound state solutions for fractional Schrödinger-Poisson systems

 School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

* Corresponding author: Qi Li

Received  June 2021 Revised  September 2021 Published  February 2022 Early access  October 2021

Fund Project: The first author is supported by NSF of Shandong Province (ZR2020MA005)

In this work, we investigate a class of fractional Schrödinger - Poisson systems
 $\begin{equation*} \left\{\begin{array}{ll}(-\triangle)^s u +V(x)u+\lambda\phi u = \mu u+|u|^{p-1}u, & x\in\ \mathbb{R}^3, \\(-\triangle)^s \phi = u^2, & x\in\ \mathbb{R}^3, \end{array}\right. \end{equation*}$
where
 $s\in(\frac{3}{4}, 1)$
,
 $p\in(3, 5)$
,
 $\lambda$
is a positive parameter. By the variational method, we show that there exists
 $\delta(\lambda)>0$
such that for all
 $\mu\in[\mu_1, \mu_1+\delta(\lambda))$
, the above fractional Schrödinger -Poisson systems possess a nonnegative bound state solutions with positive energy. Here
 $\mu_1$
is the first eigenvalue of
 $(-\triangle)^s +V(x)$
.
Citation: Xinsheng Du, Qi Li, Zengqin Zhao, Gen Li. Bound state solutions for fractional Schrödinger-Poisson systems. Mathematical Foundations of Computing, 2022, 5 (1) : 57-66. doi: 10.3934/mfc.2021023
##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [2] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984. [3] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019. [4] V. Benci and D. Fortunato, Solitary waves of nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168. [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [6] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differ. Equ., 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017. [7] S.-Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.  doi: 10.1016/j.aim.2010.07.016. [8] X. Chang and Z.-Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479. [9] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [10] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional laplacian, Matematiche, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15. [11] X. Du and A. Mao, Existence and Multiple of nontrivial solutions for a class of fractional Schrödinger equations, J. Function Space, 2017 (2017), ID3793872, 7 pp. doi: 10.1155/2017/3793872. [12] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746. [13] X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156. [14] X. He and W. Zou, Multiplicity of concentrating positive solutions for Schrödinger-Poisson equations with critical growth, Nonlinear Anal., 170 (2018), 142-170.  doi: 10.1016/j.na.2018.01.001. [15] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2. [16] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 56-108.  doi: 10.1103/PhysRevE.66.056108. [17] Z. Liu and S. Guo, On ground state solutions for the Schödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.  doi: 10.1016/j.jmaa.2013.10.066. [18] A. Mao, L. Yang, A. Qian and S. Luan, Existence and concentration of solutions of Schrödinger-possion systems, Appl. Math. Lett., 68 (2017), 8-12.  doi: 10.1016/j.aml.2016.12.014. [19] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005. [20] D. Ruiz, On the Schrödinger-Poisson-Slater systems: Behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.  doi: 10.1007/s00205-010-0299-5. [21] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $R^N$, J. Math. Phys., 54 (2013), 031501.  doi: 10.1063/1.4793990. [22] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4. [23] X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.  doi: 10.1088/0951-7715/27/2/187. [24] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [25] K. Teng, Multiple solutions for a class of fractional Schrödinger equations in $R^N$, Nonlinear Anal. Real World Appl., 21 (2015), 76-86.  doi: 10.1016/j.nonrwa.2014.06.008. [26] K. Teng and X. He, Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent, Commun. Pure Appl. Anal., 15 (2016), 991-1008.  doi: 10.3934/cpaa.2016.15.991. [27] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [28] L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [2] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984. [3] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019. [4] V. Benci and D. Fortunato, Solitary waves of nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168. [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [6] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differ. Equ., 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017. [7] S.-Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.  doi: 10.1016/j.aim.2010.07.016. [8] X. Chang and Z.-Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479. [9] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [10] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional laplacian, Matematiche, 68 (2013), 201-216.  doi: 10.4418/2013.68.1.15. [11] X. Du and A. Mao, Existence and Multiple of nontrivial solutions for a class of fractional Schrödinger equations, J. Function Space, 2017 (2017), ID3793872, 7 pp. doi: 10.1155/2017/3793872. [12] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746. [13] X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156. [14] X. He and W. Zou, Multiplicity of concentrating positive solutions for Schrödinger-Poisson equations with critical growth, Nonlinear Anal., 170 (2018), 142-170.  doi: 10.1016/j.na.2018.01.001. [15] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2. [16] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 56-108.  doi: 10.1103/PhysRevE.66.056108. [17] Z. Liu and S. Guo, On ground state solutions for the Schödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.  doi: 10.1016/j.jmaa.2013.10.066. [18] A. Mao, L. Yang, A. Qian and S. Luan, Existence and concentration of solutions of Schrödinger-possion systems, Appl. Math. Lett., 68 (2017), 8-12.  doi: 10.1016/j.aml.2016.12.014. [19] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005. [20] D. Ruiz, On the Schrödinger-Poisson-Slater systems: Behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.  doi: 10.1007/s00205-010-0299-5. [21] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $R^N$, J. Math. Phys., 54 (2013), 031501.  doi: 10.1063/1.4793990. [22] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4. [23] X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.  doi: 10.1088/0951-7715/27/2/187. [24] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [25] K. Teng, Multiple solutions for a class of fractional Schrödinger equations in $R^N$, Nonlinear Anal. Real World Appl., 21 (2015), 76-86.  doi: 10.1016/j.nonrwa.2014.06.008. [26] K. Teng and X. He, Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent, Commun. Pure Appl. Anal., 15 (2016), 991-1008.  doi: 10.3934/cpaa.2016.15.991. [27] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [28] L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.
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