# American Institute of Mathematical Sciences

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
 $$$\begin{cases} -\Delta u+V(x)u+\phi u = a (x)|u|^{p-1}u,\ x\in\mathbb{R}^3\\ -\Delta \phi = u^{2},\ x\in\mathbb{R}^3, \end{cases}$$\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. Amrein, A. 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. Dohnal, M. 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. Li, S. 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. Liu, J. 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. Liu, Z. 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. Qin, J. 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. Sun, H. 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. Wang, Q. He, L. 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. Wang, L. Tian, J. 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. Wang, L. Tian, J. 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. Wang, J. Xu, F. 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. 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.

show all references

##### References:
 [1] W. Amrein, A. 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. Dohnal, M. 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. Li, S. 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. Liu, J. 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. Liu, Z. 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. Qin, J. 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. Sun, H. 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. Wang, Q. He, L. 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. Wang, L. Tian, J. 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. Wang, L. Tian, J. 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. Wang, J. Xu, F. 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. 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.
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