# American Institute of Mathematical Sciences

February  2021, 20(2): 817-834. doi: 10.3934/cpaa.2020292

## Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity

 Department of Mathematics, Guangzhou University, Guangzhou, Guangdong, 510006, China

* Corresponding author

Received  July 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

Fund Project: The first author is supported by NSF of China (11790271), Guangdong Basic and Applied basic Research Foundation(2020A1515011019), Innovation and Development Project of Guangzhou University

In this paper, we consider the existence of a ground state nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following fractional critical problem
 $\begin{equation*} \begin{cases} (a+ b\int_{\mathbb{R}^{3}}(|(-\Delta)^{\alpha/2}u|^{2})dx)(-\Delta)^{\alpha}u+V(x)u+K(x)\phi u = |u|^{2^{\ast}-2}u+ \kappa f(x,u),\\ (-\Delta)^{\beta}\phi = K(x)u^{2}, \quad x\in\mathbb{R}^{3}, \end{cases} \end{equation*}$
where
 $a, b,\kappa$
are positive parameters,
 $\alpha\in(\frac{3}{4},1),\beta\in(0,1)$
, and
 $2^{\ast}_{\alpha} = \frac{6}{3-2\alpha}$
,
 $(-\Delta)^{\alpha}$
stands for the fractional Laplacian. By the nodal Nehari manifold method, for each
 $b>0$
, we obtain a ground state nodal solution
 $u_{b}$
and a ground-state solution
 $v_b$
to this problem when
 $\kappa\gg 1$
, where the nonlinear function
 $f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R}$
is a Carathéodory function. We also give an analysis on the behavior of
 $u_{b}$
as the parameter
 $b\to 0$
.
Citation: Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292
##### References:
 [1] T. D'Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differ. Equ., 25 (2006), 105-137.  doi: 10.1007/s00526-005-0342-9. [2] 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. [3] G. F. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math., 3 (1945), 157-165.  doi: 10.1090/qam/12351. [4] K. Cheng and Q. Gao, Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in $\mathbb{R}^{N}$, Acta Math. Sci., 38B (2018), 1712-1732.  doi: 10.1016/S0252-9602(18)30841-5. [5] S. Chen, X. Tang and F. Liao, Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions, Nonlinear Differ. Equ. Appl., 25 (2018), 23pp. doi: 10.1007/s00030-018-0531-9. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [7] Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^{3}$, J. Funct. Anal., 269 (2015), 3500-3527.  doi: 10.1016/j.jfa.2015.09.012. [8] M. F. Furtado, L. A. Maia and E. S. Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud., 8 (2008), 353-373.  doi: 10.1515/ans-2008-0207. [9] C. Ji, Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger-Poisson system in $\mathbb{R}^{3}$, Ann. Mat. Pura Appl., 198 (2019), 1563-1579.  doi: 10.1007/s10231-019-00831-2. [10] Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differ. Equ., 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006. [11] 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, 28pp. doi: 10.1142/S0219199714500369. [12] F. Li, Z. Song and Q. Zhang, Existence and uniqueness results for Kirchhoff-Schrödinger-Poisson system with general singularity, Appl. Anal., 96 (2017), 2906-2916.  doi: 10.1080/00036811.2016.1253065. [13] H. Luo, X. Tang and Z. Gao, Ground state sign-changing solutions for fractional Kirchhoff equations in bounded domains, J. Math. Phys., 59 (2018), 031504, 15 pp. doi: 10.1063/1.5026674. [14] M. Furtado, A. Maia Liliane and E. Medeiros, Least energy radial sign-changing solution for the Schrödinger-Poisson system in $\mathbb{R}^{3}$ under an asymptotically cubic nonlinearity, J. Math. Anal. Appl., 474 (2019), 544-571.  doi: 10.1016/j.jmaa.2019.01.063. [15] 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. [16] D. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Am., 32 (1960), 1529-1538.  doi: 10.1121/1.1907948. [17] 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. [18] D. Ruiz, On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.  doi: 10.1007/s00205-010-0299-5. [19] J. Sun and S. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differ. Equ., 260 (2016), 2119-2149.  doi: 10.1016/j.jde.2015.09.057. [20] K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differ. Equ., 261 (2016), 3061-3106.  doi: 10.1016/j.jde.2016.05.022. [21] D. Wang, Least energy sign-changing solutions of Kirchhoff-type equation with critical growth, J. Math. Phys., 61 (2020), 011501, 19 pp. doi: 10.1063/1.5074163. [22] D. Wang, H. Zhang and W. Guan, Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth, J. Math. Anal. Appl., 479 (2019), 2284-2301.  doi: 10.1016/j.jmaa.2019.07.052. [23] Z. Wang and H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^{3}$, Calc. Var. Partial Differ. Equ., 52 (2015), 927-943.  doi: 10.1007/s00526-014-0738-5. [24] T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differ. Equ., 27 (2006), 421-437.  doi: 10.1007/s00526-006-0015-3. [25] M. Willem, Minimax Theorems, Birkhäuser, Bosten, 1996. doi: 10.1007/978-1-4612-4146-1. [26] C. Ye and K. Teng, Ground state and sign-changing solutions for fractional Schrödinger-Poisson system with critical growth, Complex Var. Ellip. Equ., 65 (2020), 1360-1393.  doi: 10.1080/17476933.2019.1652278. [27] J. Zhang, J. M. do Ó and M. Squassina, Schrödinger-Poisson systems with a general critical nonlinearity, Commun. Contemp. Math., 19 (2017), 1650028, 16pp. doi: 10.1142/S0219199716500280. [28] J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032. [29] G. Zhao, X. Zhu and Y. Li, Existence of infinitely many solutions to a class of Kirchhoff-Schrödinger-Poisson system, Appl. Math. Comput., 256 (2015), 572-581.  doi: 10.1016/j.amc.2015.01.038.

show all references

##### References:
 [1] T. D'Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differ. Equ., 25 (2006), 105-137.  doi: 10.1007/s00526-005-0342-9. [2] 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. [3] G. F. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math., 3 (1945), 157-165.  doi: 10.1090/qam/12351. [4] K. Cheng and Q. Gao, Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in $\mathbb{R}^{N}$, Acta Math. Sci., 38B (2018), 1712-1732.  doi: 10.1016/S0252-9602(18)30841-5. [5] S. Chen, X. Tang and F. Liao, Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions, Nonlinear Differ. Equ. Appl., 25 (2018), 23pp. doi: 10.1007/s00030-018-0531-9. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [7] Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^{3}$, J. Funct. Anal., 269 (2015), 3500-3527.  doi: 10.1016/j.jfa.2015.09.012. [8] M. F. Furtado, L. A. Maia and E. S. Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud., 8 (2008), 353-373.  doi: 10.1515/ans-2008-0207. [9] C. Ji, Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger-Poisson system in $\mathbb{R}^{3}$, Ann. Mat. Pura Appl., 198 (2019), 1563-1579.  doi: 10.1007/s10231-019-00831-2. [10] Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differ. Equ., 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006. [11] 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, 28pp. doi: 10.1142/S0219199714500369. [12] F. Li, Z. Song and Q. Zhang, Existence and uniqueness results for Kirchhoff-Schrödinger-Poisson system with general singularity, Appl. Anal., 96 (2017), 2906-2916.  doi: 10.1080/00036811.2016.1253065. [13] H. Luo, X. Tang and Z. Gao, Ground state sign-changing solutions for fractional Kirchhoff equations in bounded domains, J. Math. Phys., 59 (2018), 031504, 15 pp. doi: 10.1063/1.5026674. [14] M. Furtado, A. Maia Liliane and E. Medeiros, Least energy radial sign-changing solution for the Schrödinger-Poisson system in $\mathbb{R}^{3}$ under an asymptotically cubic nonlinearity, J. Math. Anal. Appl., 474 (2019), 544-571.  doi: 10.1016/j.jmaa.2019.01.063. [15] 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. [16] D. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Am., 32 (1960), 1529-1538.  doi: 10.1121/1.1907948. [17] 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. [18] D. Ruiz, On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.  doi: 10.1007/s00205-010-0299-5. [19] J. Sun and S. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differ. Equ., 260 (2016), 2119-2149.  doi: 10.1016/j.jde.2015.09.057. [20] K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differ. Equ., 261 (2016), 3061-3106.  doi: 10.1016/j.jde.2016.05.022. [21] D. Wang, Least energy sign-changing solutions of Kirchhoff-type equation with critical growth, J. Math. Phys., 61 (2020), 011501, 19 pp. doi: 10.1063/1.5074163. [22] D. Wang, H. Zhang and W. Guan, Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth, J. Math. Anal. Appl., 479 (2019), 2284-2301.  doi: 10.1016/j.jmaa.2019.07.052. [23] Z. Wang and H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^{3}$, Calc. Var. Partial Differ. Equ., 52 (2015), 927-943.  doi: 10.1007/s00526-014-0738-5. [24] T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differ. Equ., 27 (2006), 421-437.  doi: 10.1007/s00526-006-0015-3. [25] M. Willem, Minimax Theorems, Birkhäuser, Bosten, 1996. doi: 10.1007/978-1-4612-4146-1. [26] C. Ye and K. Teng, Ground state and sign-changing solutions for fractional Schrödinger-Poisson system with critical growth, Complex Var. Ellip. Equ., 65 (2020), 1360-1393.  doi: 10.1080/17476933.2019.1652278. [27] J. Zhang, J. M. do Ó and M. Squassina, Schrödinger-Poisson systems with a general critical nonlinearity, Commun. Contemp. Math., 19 (2017), 1650028, 16pp. doi: 10.1142/S0219199716500280. [28] J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 428 (2015), 387-404.  doi: 10.1016/j.jmaa.2015.03.032. [29] G. Zhao, X. Zhu and Y. Li, Existence of infinitely many solutions to a class of Kirchhoff-Schrödinger-Poisson system, Appl. Math. Comput., 256 (2015), 572-581.  doi: 10.1016/j.amc.2015.01.038.
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