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April  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 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.
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