Non-autonomous Schrödinger-Poisson system in <inline-formula><tex-math id="M1">\begin{document}  $\mathbb{R}^{3}$\end{document}</tex-math></inline-formula>

Juntao Sun; Tsung-Fang Wu; Zhaosheng Feng

doi:10.3934/dcds.2018077

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 and Continuous Dynamical Systems, 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 94 (2004), 1-31.
[13]	T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
[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.
[15]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 94 (2004), 1-31.
[13]	T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
[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.
[15]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[31]	G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x.
[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.
[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.
[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.

[1]	Jin-Cai Kang, Xiao-Qi Liu, Chun-Lei Tang. Ground state sign-changing solution for Schrödinger-Poisson system with steep potential well. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022112
[2]	Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064
[3]	Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2805-2816. doi: 10.3934/jimo.2020095
[4]	Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
[5]	Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257
[6]	Rong Cheng, Jun Wang. Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021317
[7]	Zhengping Wang, Huan-Song Zhou. Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 809-816. doi: 10.3934/dcds.2007.18.809
[8]	Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[9]	Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
[10]	Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214
[11]	Sitong Chen, Wennian Huang, Xianhua Tang. Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3055-3066. doi: 10.3934/dcdss.2020339
[12]	Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure and Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048
[13]	Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447
[14]	Kaimin Teng, Xian Wu. Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1157-1187. doi: 10.3934/cpaa.2022014
[15]	Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
[16]	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 and Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103
[17]	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 and Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126
[18]	Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1737-1754. doi: 10.3934/cpaa.2021039
[19]	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 and Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079
[20]	Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors