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doi: 10.3934/dcdsb.2021134

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 Published  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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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

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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

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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

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V. Benci and D. Fortunato, Variational Methods in Nonlinear Filed Equations, Springer, Cham, 2014. doi: 10.1007/978-3-319-06914-2.  Google Scholar

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R. BenguriaH. 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

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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

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J. ChenX. 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

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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

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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

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[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. FortunatoL. 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. IllnerO. 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. IllnerH. LangeB. 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. LiuX.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. LiuJ.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. LiuZ.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

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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

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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

show all references

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. BenguriaH. 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. ChenX. 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. DingL. LiY.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. FortunatoL. 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. IllnerO. 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. IllnerH. LangeB. 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. LiuX.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. LiuJ.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. LiuZ.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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Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1497-1519. doi: 10.3934/cpaa.2021030

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