doi: 10.3934/dcdsb.2021317
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Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials

1. 

Department of Mathematics, Nanjing University of Information Science & Technology, Nanjing, Jiangsu, 210044, China

2. 

Institute of Applied System Analysis, Jiangsu University, Zhenjiang, Jiangsu, 212013, China

*Corresponding author: Jun Wang

Received  September 2021 Revised  November 2021 Early access January 2022

Fund Project: This work was supported by NNSF of China(Grants 11971202), Outstanding Young foundation of Jiangsu Province No. BK20200042 and the Six big talent peaks project in Jiangsu Province(XYDXX-015)

In the present paper we study a class of Schrödinger-Poisson equations
$\begin{equation} \begin{cases} -\Delta u+V(x)u+\phi u = a (x)|u|^{p-1}u,\ x\in\mathbb{R}^3\\ -\Delta \phi = u^{2},\ x\in\mathbb{R}^3, \end{cases} \end{equation}\quad\quad\quad (1)$
where
$ V(x) $
and
$ a(x) $
are of different forms on the half space, i.e.
$ V(x) = V_{1}(x), a(x) = a_{1}(x) $
for
$ x_{1}>0 $
and
$ V(x) = V_{2}(x), a(x) = a_{2}(x) $
for
$ x_{1}<0 $
, where
$ V_{1},V_{2},a_{1} $
and
$ a_{2} $
are periodic in each coordinate direction. By using a concentration compactness discussion, we establish the existence of surface gap soliton ground state of (1) for
$ p\in [3,5) $
. We also give a Mountain-Pass type ground state of (1) for
$ p\in (3,5) $
.
Citation: Rong Cheng, Jun Wang. Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021317
References:
[1]

W. AmreinA. Berthier and V. Georgescu, $L^{p}$-inequalities for the Laplacian and unique continuation, Ann. Inst. Fourier (Grenoble), 31 (1981), 153-168.  doi: 10.5802/aif.843.

[2]

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.

[3]

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.

[4]

Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal., 251 (2007), 546-572.  doi: 10.1016/j.jfa.2007.07.005.

[5]

T. DohnalM. Plum and W. Reichel, Surface gap soliton ground states for the nonlinear Schrödinger equation, Comm. Math. Phys., 308 (2011), 511-542.  doi: 10.1007/s00220-011-1320-z.

[6]

M. Du, L. Tian, J. Wang and F. Zhang, Existence and asymptotic behavior of solutions for nonlinear Schrödinger-Poisson systems with steep potential well, J. Math. Phys., 57 (2016), 031502, 19 pp. doi: 10.1063/1.4941036.

[7]

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

[8]

X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889.  doi: 10.1007/s00033-011-0120-9.

[9]

X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156.

[10]

I. Ianni and D. Ruiz, Ground and bound states for a static Schrödinger-Poisson-Slater problem, Commun. Contemp. Math., 14 (2012), 1250003, 22 pp. doi: 10.1142/S0219199712500034.

[11]

Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006.

[12]

G. Li, S. Peng and C. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 52 (2011), 053505, 19 pp. doi: 10.1063/1.3585657.

[13]

G. LiS. Peng and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.  doi: 10.1142/S0219199710004068.

[14]

E.-H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[15]

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

[16]

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

[17]

Z. LiuJ. Su and Z. Wang, Solutions of elliptic problems with nonlinearities of linear growth, Calc. Var. Partial Differential Equations, 35 (2009), 463-480.  doi: 10.1007/s00526-008-0215-0.

[18]

Z. LiuZ. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.  doi: 10.1007/s10231-015-0489-8.

[19]

F.-Y. QinJ. Wang and J. Yang, Infinitely many positive solutions for Schrödinger-Poisson systems with nonsymmetry potentials, Discrete Contin. Dyn. Syst., 41 (2021), 4705-4736.  doi: 10.3934/dcds.2021054.

[20]

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.

[21]

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.

[22]

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.

[23]

M. Schechter and B. Simon, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl., 77 (1980), 482-492.  doi: 10.1016/0022-247X(80)90242-5.

[24]

M. Struwe, Variational Methods, 4$^{nd}$ edition, Springer-Verlag, Berlin, 2008. doi: 978-3-540-74012-4.

[25]

J. SunH. Chen and Juan 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.

[26]

J. Sun and S. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149.  doi: 10.1016/j.jde.2015.09.057.

[27]

J. WangQ. HeL. Xiao and F. Zhang, Positive solutions for Schrödinger system with asymptotically periodic potentials, Nonlinear Anal., 134 (2016), 215-235.  doi: 10.1016/j.na.2016.01.011.

[28]

J. WangL. TianJ. Xu and F. Zhang, Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth, Z. Angew. Math. Phys., 66 (2015), 2441-2471.  doi: 10.1007/s00033-015-0531-0.

[29]

J. WangL. TianJ. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in ${\mathbb{R}}^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273.  doi: 10.1007/s00526-012-0548-6.

[30]

J. WangJ. XuF. Zhang and X. Chen, Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, 26 (2013), 1377-1399.  doi: 10.1088/0951-7715/26/5/1377.

[31]

Z. Wang and H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in ${\mathbb{R}}^3$, Calc. Var. Partial Differential Equations, 52 (2015), 927-943.  doi: 10.1007/s00526-014-0738-5.

[32]

Z. Wang and H. Zhou, Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Eur. Math. Soc., 11 (2009), 545-573.  doi: 10.4171/JEMS/160.

[33]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[34]

L. ZhaoH. 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.

show all references

References:
[1]

W. AmreinA. Berthier and V. Georgescu, $L^{p}$-inequalities for the Laplacian and unique continuation, Ann. Inst. Fourier (Grenoble), 31 (1981), 153-168.  doi: 10.5802/aif.843.

[2]

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.

[3]

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.

[4]

Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal., 251 (2007), 546-572.  doi: 10.1016/j.jfa.2007.07.005.

[5]

T. DohnalM. Plum and W. Reichel, Surface gap soliton ground states for the nonlinear Schrödinger equation, Comm. Math. Phys., 308 (2011), 511-542.  doi: 10.1007/s00220-011-1320-z.

[6]

M. Du, L. Tian, J. Wang and F. Zhang, Existence and asymptotic behavior of solutions for nonlinear Schrödinger-Poisson systems with steep potential well, J. Math. Phys., 57 (2016), 031502, 19 pp. doi: 10.1063/1.4941036.

[7]

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

[8]

X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889.  doi: 10.1007/s00033-011-0120-9.

[9]

X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156.

[10]

I. Ianni and D. Ruiz, Ground and bound states for a static Schrödinger-Poisson-Slater problem, Commun. Contemp. Math., 14 (2012), 1250003, 22 pp. doi: 10.1142/S0219199712500034.

[11]

Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006.

[12]

G. Li, S. Peng and C. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 52 (2011), 053505, 19 pp. doi: 10.1063/1.3585657.

[13]

G. LiS. Peng and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.  doi: 10.1142/S0219199710004068.

[14]

E.-H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[15]

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

[16]

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

[17]

Z. LiuJ. Su and Z. Wang, Solutions of elliptic problems with nonlinearities of linear growth, Calc. Var. Partial Differential Equations, 35 (2009), 463-480.  doi: 10.1007/s00526-008-0215-0.

[18]

Z. LiuZ. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.  doi: 10.1007/s10231-015-0489-8.

[19]

F.-Y. QinJ. Wang and J. Yang, Infinitely many positive solutions for Schrödinger-Poisson systems with nonsymmetry potentials, Discrete Contin. Dyn. Syst., 41 (2021), 4705-4736.  doi: 10.3934/dcds.2021054.

[20]

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.

[21]

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.

[22]

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.

[23]

M. Schechter and B. Simon, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl., 77 (1980), 482-492.  doi: 10.1016/0022-247X(80)90242-5.

[24]

M. Struwe, Variational Methods, 4$^{nd}$ edition, Springer-Verlag, Berlin, 2008. doi: 978-3-540-74012-4.

[25]

J. SunH. Chen and Juan 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.

[26]

J. Sun and S. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149.  doi: 10.1016/j.jde.2015.09.057.

[27]

J. WangQ. HeL. Xiao and F. Zhang, Positive solutions for Schrödinger system with asymptotically periodic potentials, Nonlinear Anal., 134 (2016), 215-235.  doi: 10.1016/j.na.2016.01.011.

[28]

J. WangL. TianJ. Xu and F. Zhang, Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth, Z. Angew. Math. Phys., 66 (2015), 2441-2471.  doi: 10.1007/s00033-015-0531-0.

[29]

J. WangL. TianJ. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in ${\mathbb{R}}^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273.  doi: 10.1007/s00526-012-0548-6.

[30]

J. WangJ. XuF. Zhang and X. Chen, Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, 26 (2013), 1377-1399.  doi: 10.1088/0951-7715/26/5/1377.

[31]

Z. Wang and H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in ${\mathbb{R}}^3$, Calc. Var. Partial Differential Equations, 52 (2015), 927-943.  doi: 10.1007/s00526-014-0738-5.

[32]

Z. Wang and H. Zhou, Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Eur. Math. Soc., 11 (2009), 545-573.  doi: 10.4171/JEMS/160.

[33]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[34]

L. ZhaoH. 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.

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