# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021134
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## Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms

 College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

* Corresponding author: Xueqin Peng

Received  January 2021 Revised  March 2021 Early access April 2021

Fund Project: This research is supported by the National Natural Science Foundation of China, Grant No. 11171220

In this paper, we consider the following Schrödinger-Poisson system with double quasi-linear terms
 $\begin{equation*} \label{1.1} \begin{cases} -\Delta u+V(x)u+\phi u-\frac{1}{2}u\Delta u^2 = \lambda f(x,u),\; &\; {\rm{in}}\; \mathbb{R}^{3},\\ -\triangle\phi-\varepsilon^4\Delta_4\phi = u^{2},\; &\; {\rm{in}}\; \mathbb{R}^{3},\\ \end{cases} \end{equation*}$
where
 $\lambda,\varepsilon$
are positive parameters. Under suitable assumptions on
 $V$
and
 $f$
, we prove that the above system admits at least one pair of positive solutions for
 $\lambda$
large by using perturbation method and truncation technique. Furthermore, we research the asymptotical behavior of solutions with respect to the parameters
 $\lambda$
and
 $\varepsilon$
respectively. These results extend and improve some existing results in the literature.
Citation: Xueqin Peng, Gao Jia. Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021134
##### References:
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Equ., 52 (2015), 565-586.  doi: 10.1007/s00526-014-0724-y.  Google Scholar [28] J.Q. Liu and Z.Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differential Equations, 257 (2014), 2874-2899.  doi: 10.1016/j.jde.2014.06.002.  Google Scholar [29] X.Q. Liu, J.Q. Liu and Z.Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar [30] Z.L. Liu and J.X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299.  doi: 10.1006/jdeq.2000.3867.  Google Scholar [31] Z.L. Liu, Z.Q. Wang and J.J. Zhang, Infinitely many sign-changing solutions for nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.  doi: 10.1007/s10231-015-0489-8.  Google Scholar [32] P.A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, Wien, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar [33] J.J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for a class of modified Schrödinger-Poisson systems, J. Math. Anal. Appl., 408 (2013), 713-724.  doi: 10.1016/j.jmaa.2013.06.011.  Google Scholar [34] P.H. Rabinowitz, Minimax methods in critical points theory with application to differential equations, CBMS Regional Conf. Ser. Math. vol. 65. Am. Math. Soc. Providence, 1986. doi: 10.1090/cbms/065.  Google Scholar [35] 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 [36] W. Shuai and Q. Wang, Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$, Z. Angew. Math. Phys., 66 (2015), 3267-3282.  doi: 10.1007/s00033-015-0571-5.  Google Scholar [37] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 2 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.  Google Scholar [38] Z.P. Wang and H.S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2012), 809-816.  doi: 10.3934/dcds.2007.18.809.  Google Scholar [39] Z.P. Wang and H.S. 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 [40] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [41] L.G. Zhao and F.K. 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.  Google Scholar [42] L.G. Zhao and F.K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.  doi: 10.1016/j.na.2008.02.116.  Google Scholar

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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.  Google Scholar [2] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Comm. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.  Google Scholar [3] T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. R. Soc. Edinburgh Sect. A., 134 (2004), 893-906.  doi: 10.1017/S030821050000353X.  Google Scholar [4] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.  doi: 10.1016/j.jmaa.2008.03.057.  Google Scholar [5] Z. Ba and X. He, Solutions for a class of Schrödinger-Poisson system in bounded domains, J. Appl. Math. Comput., 51 (2016), 287-297.  doi: 10.1007/s12190-015-0905-7.  Google Scholar [6] 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.  Google Scholar [7] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168.  Google Scholar [8] V. Benci and D. Fortunato, Variational Methods in Nonlinear Filed Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-06914-2.  Google Scholar [9] R. Benguria, H. Brézis and E.H. Lieb, The Thomas-Fermi-von Weizsacker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.  doi: 10.1007/BF01942059.  Google Scholar [10] K. Benmilh and O. Kavian, Existence and asymptotical behavior of standing waves for quasilinear Schrödinger-Poisson systems in $\mathbb{R}^3$, Ann. I. H. Poincaré Anal. Non Linéaire, 25 (2008), 449-470.  doi: 10.1016/j.anihpc.2007.02.002.  Google Scholar [11] I. Catto and P.L. Lions, Binding of atoms and stability of molecules in Hartree and ThomasFermi type theories. PART 1: A necessary and sufficient condition for the stability of generalmolecular system, Comm. Partial Differ. Equ., 17 (1992), 1051-1110.  doi: 10.1080/03605309208820878.  Google Scholar [12] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.  Google Scholar [13] J. Chen, X. Tang and Z. Gao, Existence of multiple solutions for modified Schrödinger-Kirchhoff-Poisson type systems via perturbation method with sign-changing potential, Comp. Math. Appl., 73 (2017), 505-519.  doi: 10.1016/j.camwa.2016.12.006.  Google Scholar [14] L. Chen, X. Feng and X. Hao, The existence of sign-changing solution for a class of quasilinear Schrödinger-Poisson systems via perturbation method, Bound. Value Probl., 2019 (2019), Paper No. 159, 19 pp. doi: 10.1186/s13661-019-1272-3.  Google Scholar [15] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar [16] D. G. Costa, On a class of Elliptic systems in $\mathbb{R}^N$, Electron. J. Differential Equations, 1994 (1994), No. 7, 1–14.  Google Scholar [17] L. Ding, L. Li, Y.J. Meng and C.L. Zhuang, Existence and asymptotic behavior of ground state solution for quasi-linear Schrödinger-Poisson system in $\mathbb{R}^3$, Topol. Methods Nonlinear Anal., 47 (2016), 241-264.   Google Scholar [18] Y. Ding and A. Szulkin, Bounded states for semilinear Schrödinger equation with sign-changing potential, Calc. Var. Partial Differ. Equ., 29 (2007), 397-419.  doi: 10.1007/s00526-006-0071-8.  Google Scholar [19] X. Feng and Y. Zhang, Existence of non-trivial solution for a class of modified Schrödinger-Poisson equations via perturbation method, J. Math. Anal. Appl., 442 (2016), 673-684.  doi: 10.1016/j.jmaa.2016.05.002.  Google Scholar [20] G.M. Figueiredo and G. Siciliano, Existence and asymptotical behavior of solutions for Schrödinger-Poisson system with a critical nonlinearity, Z. Angew. Math. Phys., 71 (2020), 130. doi: 10.1007/s00033-020-01356-y.  Google Scholar [21] G.M. Figueiredo and G. Siciliano, Quasi-linear Schrödinger-Poisson system under an exponential critical nonlinearity: Existence and asymptotic behavior of solutions, Arch. Math., 112 (2019), 313-327.  doi: 10.1007/s00013-018-1287-5.  Google Scholar [22] D. Fortunato, L. Orsina and L. Pisani, Born-Infeld type equations for the electrostatic fields, J. Math. Phys., 43 (2002), 5698-5706.  doi: 10.1063/1.1508433.  Google Scholar [23] L. Gu, H. Jin and J. Zhang, Sign-changing solutions for nonlinear Schrödinger-Poisson systems with subquadratic or quadratic growth at infinity, Nonlinear Anal., 198 (2020), 111897, 16 pp. doi: 10.1016/j.na.2020.111897.  Google Scholar [24] R. Illner, O. Kavian and H. Lange, Stationary solutions of quasi-linear Schrödinger-Poisson systems, J. Differential Equations, 145 (1998), 1-16.  doi: 10.1006/jdeq.1997.3405.  Google Scholar [25] R. Illner, H. Lange, B. Toomire and P. Zweifel, On quasi-linear Schrödinger-Poisson systems, Math. Methods Appl. Sci., 20 (1997), 1223-1238.  doi: 10.1002/(SICI)1099-1476(19970925)20:14<1223::AID-MMA911>3.0.CO;2-O.  Google Scholar [26] B. Li and H. Yang, The modified quantum Wigner system in weighted $L^2$-space, Bull. Aust. Math. Soc., 95 (2017), 73-83.  doi: 10.1017/S0004972716000666.  Google Scholar [27] J.Q. Liu, X.Q. Liu and Z.Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger-Poisson system, Calc. Var. Partial Differ. Equ., 52 (2015), 565-586.  doi: 10.1007/s00526-014-0724-y.  Google Scholar [28] J.Q. Liu and Z.Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differential Equations, 257 (2014), 2874-2899.  doi: 10.1016/j.jde.2014.06.002.  Google Scholar [29] X.Q. Liu, J.Q. Liu and Z.Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar [30] Z.L. Liu and J.X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299.  doi: 10.1006/jdeq.2000.3867.  Google Scholar [31] Z.L. Liu, Z.Q. Wang and J.J. Zhang, Infinitely many sign-changing solutions for nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.  doi: 10.1007/s10231-015-0489-8.  Google Scholar [32] P.A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, Wien, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar [33] J.J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for a class of modified Schrödinger-Poisson systems, J. Math. Anal. Appl., 408 (2013), 713-724.  doi: 10.1016/j.jmaa.2013.06.011.  Google Scholar [34] P.H. Rabinowitz, Minimax methods in critical points theory with application to differential equations, CBMS Regional Conf. Ser. Math. vol. 65. Am. Math. Soc. Providence, 1986. doi: 10.1090/cbms/065.  Google Scholar [35] 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 [36] W. Shuai and Q. Wang, Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$, Z. Angew. Math. Phys., 66 (2015), 3267-3282.  doi: 10.1007/s00033-015-0571-5.  Google Scholar [37] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 2 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.  Google Scholar [38] Z.P. Wang and H.S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2012), 809-816.  doi: 10.3934/dcds.2007.18.809.  Google Scholar [39] Z.P. Wang and H.S. 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 [40] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [41] L.G. Zhao and F.K. 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.  Google Scholar [42] L.G. Zhao and F.K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.  doi: 10.1016/j.na.2008.02.116.  Google Scholar
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