# 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  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 & Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292
##### References:
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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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] D. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Am., 32 (1960), 1529-1538.  doi: 10.1121/1.1907948.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] M. Willem, Minimax Theorems, Birkhäuser, Bosten, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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