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2018, 38(4): 1889-1933. doi: 10.3934/dcds.2018077

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

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
$3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$
.
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.

[2]

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

[3]

A. Azzollini, 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, 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, Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations, 5 (1997), 1-11.

[6]

H. Brézis, 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, 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, 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, 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, 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, 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, V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 94 (2004), 1-31.

[13]

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

[14]

P. Drábek, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 39-404. doi: 10.1142/S021919970800282X.

[3]

A. Azzollini, 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, 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, Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations, 5 (1997), 1-11.

[6]

H. Brézis, 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, 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, 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, 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, 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, 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, V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 94 (2004), 1-31.

[13]

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

[14]

P. Drábek, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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.

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