American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity
  • DCDS Home
  • This Issue
  • Next Article
    Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities
April  2018, 38(4): 1889-1933. doi: 10.3934/dcds.2018077

Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$

Juntao Sun 1,2,, , Tsung-Fang Wu 3, and  Zhaosheng Feng 4,

1. 

School of Mathematics and Statistics, Shandong University of Technology Zibo 255049, China
2. 

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
3. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
4. 

School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, Texas 78539, USA

* Corresponding author

Received  November 2016 Revised  October 2017 Published  January 2018

We study the existence of positive solutions for the non-autonomous Schrödinger-Poisson system:
$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$
where
$\lambda >0$
,
$2 < p \le 4$
 and both
$K\left( x\right) $
 and
$a\left( x\right) $
 are nonnegative functions in
$\mathbb{R}^{3}$
, which satisfy the given conditions, but not require any symmetry property. Assuming that
$% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$
 and
$\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$
, we explore the existence of positive solutions, depending on the parameters
$\lambda$
 and
$p$
. More importantly, we establish the existence of ground state solutions in the case of
$3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$
.
Keywords: Positive solution, Schrödinger-Poisson system, variational method, ground state, fibering maps, Sobolev embedding theorem, concentration-compactness principle.
Mathematics Subject Classification: Primary: 35J20, 35J61, 35A01; Secondary: 35B40.
Citation: Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng. Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1889-1933. doi: 10.3934/dcds.2018077
References:
[1]

A. Ambrosetti, On the Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.  Google Scholar

[2]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 39-404.  doi: 10.1142/S021919970800282X.  Google Scholar

[3]

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

[4]

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

[5]

P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations, 5 (1997), 1-11.   Google Scholar

[6]

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer Math. Soc., 88 (1983), 486-490.   Google Scholar

[7]

K. J. Brown and T. F. Wu, A fibrering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations, 69 (2007), 1-9.   Google Scholar

[8]

K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications, Differential Integral Equations, 22 (2009), 1097-1114.   Google Scholar

[9]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[10]

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

[11]

C. Y. Chen, Y. C. Kuo and T. F. Wu, Existence and multiplicity of positive solutions for the nonlinear Schrödinger-Poisson equations, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 745-764.  doi: 10.1017/S0308210511000692.  Google Scholar

[12]

G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 94 (2004), 1-31.   Google Scholar

[13]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.   Google Scholar

[14]

P. Drábek and S. I. Pohozaev, Positive solutions for the $p$ -Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787.  Google Scholar

[15]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[16]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595.   Google Scholar

[17]

I. Ianni and G. Vaira, Non-radial sign-changing solutions for the Schrödinger-Poisson problem in the semiclassical limit, Nonlinear Differ. Equ. Appl., 22 (2015), 741-776.  doi: 10.1007/s00030-014-0303-0.  Google Scholar

[18]

M. K. Kwong, Uniqueness of positive solution of $Δ u-u+u^{p}=0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.   Google Scholar

[19]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case Ⅰ, Ann. Inst. H. Poincar é Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[20]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case Ⅱ, Ann. Inst. H. Poincar é Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[21]

A. Mao, L. Yang, A. Qian and S. Luan, Existence and concentration of solutions of Schrödinger-Poisson system, Applied Mathematics Letters, 68 (2017), 8-12.  doi: 10.1016/j.aml.2016.12.014.  Google Scholar

[22]

Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.  doi: 10.1090/S0002-9947-1960-0111898-8.  Google Scholar

[23]

W. M. Ni and I. Takagi, On the shape of least energy solution to a Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[24]

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

[25]

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

[26]

O. Sánchez and J. Soler, Long-time dynamics of the Schrödinger-Poisson-Slater system, J. Statist. Phys., 114 (2004), 179-204.  doi: 10.1023/B:JOSS.0000003109.97208.53.  Google Scholar

[27]

J. Sun, H. Chen and J. J. Nieto, On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 252 (2012), 3365-3380.  doi: 10.1016/j.jde.2011.12.007.  Google Scholar

[28]

J. Sun and T. F. Wu, On the nonlinear Schrödinger-Poisson systems with sign-changing potential, Z. Angew. Math. Phys., 66 (2015), 1649-1669.  doi: 10.1007/s00033-015-0494-1.  Google Scholar

[29]

J. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Differential Equations, 260 (2016), 586-627.  doi: 10.1016/j.jde.2015.09.002.  Google Scholar

[30]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar

[31]

G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.  Google Scholar

[32]

Z. Wang and H. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.  doi: 10.3934/dcds.2007.18.809.  Google Scholar

[33]

L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential Equations, 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.  Google Scholar

[34]

L. Zhao and F. Zhao, On the existence of solutions for the Schr ödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.  Google Scholar

show all references

References:
[1]

A. Ambrosetti, On the Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.  Google Scholar

[2]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 39-404.  doi: 10.1142/S021919970800282X.  Google Scholar

[3]

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

[4]

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

[5]

P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations, 5 (1997), 1-11.   Google Scholar

[6]

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer Math. Soc., 88 (1983), 486-490.   Google Scholar

[7]

K. J. Brown and T. F. Wu, A fibrering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations, 69 (2007), 1-9.   Google Scholar

[8]

K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications, Differential Integral Equations, 22 (2009), 1097-1114.   Google Scholar

[9]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[10]

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

[11]

C. Y. Chen, Y. C. Kuo and T. F. Wu, Existence and multiplicity of positive solutions for the nonlinear Schrödinger-Poisson equations, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 745-764.  doi: 10.1017/S0308210511000692.  Google Scholar

[12]

G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 94 (2004), 1-31.   Google Scholar

[13]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.   Google Scholar

[14]

P. Drábek and S. I. Pohozaev, Positive solutions for the $p$ -Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787.  Google Scholar

[15]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[16]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595.   Google Scholar

[17]

I. Ianni and G. Vaira, Non-radial sign-changing solutions for the Schrödinger-Poisson problem in the semiclassical limit, Nonlinear Differ. Equ. Appl., 22 (2015), 741-776.  doi: 10.1007/s00030-014-0303-0.  Google Scholar

[18]

M. K. Kwong, Uniqueness of positive solution of $Δ u-u+u^{p}=0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.   Google Scholar

[19]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case Ⅰ, Ann. Inst. H. Poincar é Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[20]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case Ⅱ, Ann. Inst. H. Poincar é Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[21]

A. Mao, L. Yang, A. Qian and S. Luan, Existence and concentration of solutions of Schrödinger-Poisson system, Applied Mathematics Letters, 68 (2017), 8-12.  doi: 10.1016/j.aml.2016.12.014.  Google Scholar

[22]

Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.  doi: 10.1090/S0002-9947-1960-0111898-8.  Google Scholar

[23]

W. M. Ni and I. Takagi, On the shape of least energy solution to a Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[24]

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

[25]

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

[26]

O. Sánchez and J. Soler, Long-time dynamics of the Schrödinger-Poisson-Slater system, J. Statist. Phys., 114 (2004), 179-204.  doi: 10.1023/B:JOSS.0000003109.97208.53.  Google Scholar

[27]

J. Sun, H. Chen and J. J. Nieto, On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 252 (2012), 3365-3380.  doi: 10.1016/j.jde.2011.12.007.  Google Scholar

[28]

J. Sun and T. F. Wu, On the nonlinear Schrödinger-Poisson systems with sign-changing potential, Z. Angew. Math. Phys., 66 (2015), 1649-1669.  doi: 10.1007/s00033-015-0494-1.  Google Scholar

[29]

J. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Differential Equations, 260 (2016), 586-627.  doi: 10.1016/j.jde.2015.09.002.  Google Scholar

[30]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar

[31]

G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.  Google Scholar

[32]

Z. Wang and H. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.  doi: 10.3934/dcds.2007.18.809.  Google Scholar

[33]

L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential Equations, 255 (2013), 1-23.  doi: 10.1016/j.jde.2013.03.005.  Google Scholar

[34]

L. Zhao and F. Zhao, On the existence of solutions for the Schr ödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.  Google Scholar

[1]

Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
[2]

Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257
[3]

Zhengping Wang, Huan-Song Zhou. Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 809-816. doi: 10.3934/dcds.2007.18.809
[4]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[5]

Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
[6]

Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214
[7]

Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
[8]

Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048
[9]

Yi He, Lu Lu, Wei Shuai. Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103
[10]

Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126
[11]

Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079
[12]

Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058
[13]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[14]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[15]

Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266
[16]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
[17]

Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure & Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867
[18]

Dengfeng Lü. Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (2) : 605-626. doi: 10.3934/cpaa.2018033
[19]

Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427
[20]

Margherita Nolasco. Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1411-1419. doi: 10.3934/cpaa.2010.9.1411

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (222)
  • HTML views (467)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences