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doi: 10.3934/dcds.2021054

Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials

1. 

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

2. 

School of science, Jiangsu University of Science and Technology, Zhenjiang 212003, China

* Corresponding author: Jun Wang

Received  November 2020 Published  April 2021

Fund Project: This work was supported by NNSF of China (Grants 11971202, 11671077, 11601194), Outstanding Young foundation of Jiangsu Province No. BK20200042 and the Six big talent peaks project in Jiangsu Province(XYDXX-015)

The present paper deals with a class of Schrödinger-poisson system. Under some suitable assumptions on the decay rate of the coefficients, we derive the existence of infinitely many positive solutions to the problem by using purely variational methods. Comparing to the previous works, we encounter some new challenges because of nonlocal term. By doing some delicate estimates for the nonlocal term we overcome the difficulty and find infinitely many positive solutions.

Citation: Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021054
References:
[1]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Comm. Contemp. Math., 10 (2008), 1-14.  doi: 10.1142/S021919970800282X.  Google Scholar

[2]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.  Google Scholar

[3]

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[4]

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.  Google Scholar

[5]

A. AzzolliniP. d'venia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791.  doi: 10.1016/j.anihpc.2009.11.012.  Google Scholar

[6]

W. BaoN. J. Mauser and H.-P. Stimming, Effective one particle quantum dynamics of electrons: A numerical study of the Schrödinger-Poisson-X $\alpha$ model, Commun. Math. Sci., 1 (2003), 809-828.  doi: 10.4310/CMS.2003.v1.n4.a8.  Google Scholar

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V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Poission equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.  Google Scholar

[8]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168.  Google Scholar

[9]

V. Benci and D. Fortunato, Solitons in Schrödinger-Maxwell equations, J. Fixed Point Theory Appl., 15 (2014), 101-132.  doi: 10.1007/s11784-014-0184-1.  Google Scholar

[10]

H. Berestycki and P.L. Lions, Nonlinear scalar field equations II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.  Google Scholar

[11]

S. Bernstein, Sur une classe d'équations fonctionelles aux dérivées partielles, Bull. Acad. Sci. USSR Sér., 4 (1940), 17-26.,  Google Scholar

[12]

D. Bonheure and C. Mercuri, Embedding theorems and existence results for nonlinear Schrödinger-Poisson systems with unbounded and vanishing potentials, J. Diffierential Equations, 251 (2011), 1056-1085.  doi: 10.1016/j.jde.2011.04.010.  Google Scholar

[13]

I. CattoJ. DolbeaultO. Sanchez and J. Soler, Existence of steady states for the Maxwell-Schrödinger-Poisson system: Exploring the applicability of the concentration-compactness principle, Math. Models Methods Appl. Sci., 23 (2013), 1915-1938.  doi: 10.1142/S0218202513500541.  Google Scholar

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G. CeramiR. Molle and D. Passaseo, Multiplicity of positive and nodal solutions for scalar field equations, J. Diff. Equations, 257 (2014), 3554-3606.  doi: 10.1016/j.jde.2014.07.002.  Google Scholar

[15]

G. Cerami and R. Molle, Positive bound state solution for some Schrödinger-Poisson systems, Nonlinearity, 29 (2016), 3013-3119.  doi: 10.1088/0951-7715/29/10/3103.  Google Scholar

[16]

G. Cerami and R. Molle, Infinitely many positive standing waves for Schrödinger equations with competing coefficients, Comm. Partial Differential Equations, 44 (2019), 73-109.  doi: 10.1080/03605302.2018.1541905.  Google Scholar

[17]

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Diffierential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.  Google Scholar

[18]

G. CeramiD. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.  doi: 10.1002/cpa.21410.  Google Scholar

[19]

S.-M. ChangS. GustafsonK. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39 (2008), 1070-111.  doi: 10.1137/050648389.  Google Scholar

[20]

T. D'Aprile and J.-C. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differ. Equ., 25 (2006), 105-137.  doi: 10.1007/s00526-005-0342-9.  Google Scholar

[21]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon- Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.  doi: 10.1017/S030821050000353X.  Google Scholar

[22]

T. D'Aprile and D. Mugnai, Nonexistence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.  Google Scholar

[23]

M. Furtado, L. A. Maia and E. S. Medeiros, A note on the existence of a positive solution for a non-autonomous Schrödinger-Poisson system Analysis and Topology in Nonlinear Differential Equations, Progress in Nonlinear Differential Equations and their Applications, New York: Springer, 85 (2010), 277-286.  Google Scholar

[24]

B. GidasW.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[25]

X.-M. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 5 (2011), 869-889.  doi: 10.1007/s00033-011-0120-9.  Google Scholar

[26]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson system with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595.  doi: 10.1515/ans-2008-0305.  Google Scholar

[27]

Y.-S. Jiang and H.-S. Zhou, Bound states for a stationary nonlinear Schrödinger-Poisson system with signchanging potential in $\mathbb{R}^{3}$, Acta Mathematica Scientia, 29 (2009), 1095-1104.  doi: 10.1016/S0252-9602(09)60088-6.  Google Scholar

[28]

M.-K. Kwong, Uniqueness of positive solutions of $-\Delta u-u+u^p = 0$ in $\mathbb{R}^N$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[29]

E.-H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[30]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, I-II, Ann. Inst. H. Poincaré Anal, Non Lineáire, 1 (1984), 109-145/223-283. doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[31]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.  doi: 10.1007/BF01205672.  Google Scholar

[32]

C. Mercuri, Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 Mat. Appl., 19 (2008), 211-227.  doi: 10.4171/RLM/520.  Google Scholar

[33]

C. Mercuri and T.-M. Tyler, On a class of nonlinear Schrödinger-Poisson systems involving a nonradial charge density, preprint, arXiv: 1805.00964. doi: 10.4171/rmi/1158.  Google Scholar

[34]

D. Mugnai, The Schrödinger-Poisson system with positive potential, Comm. Partial Diff. Eqns., 36 (2011), 1099-1117.  doi: 10.1080/03605302.2011.558551.  Google Scholar

[35]

E.-S. Noussair and J.-C. Wei, On the effect of the domain geometry on the existence and profile of nodal solution of some singularly perturbed semilinear Dirichlet problem, Indiana Univ. Math. J., 46 (1997), 1321-1332.  doi: 10.1512/iumj.1997.46.1401.  Google Scholar

[36]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Functional Analysis, 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[37]

J. Slater, A simplification of the Hartree-Fock Method, Phys. Rev., 81 (1951), 385-390.   Google Scholar

[38]

J. SunT. Wu and Z. Feng, Non-autonomous Schrödinger-Poisson System in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933.  doi: 10.3934/dcds.2018077.  Google Scholar

[39]

G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.  Google Scholar

[40]

J. WangL. X. TianJ. X. Xu and F. B. 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.  Google Scholar

[41]

J. WangL. X. TianJ. X. Xu and F. B. Zhang, Existence and concentration of positive ground state solutions for semilinear Schrödinger-Poisson systems, Adv. Nonlinear Stud., 13 (2013), 553-582.  doi: 10.1515/ans-2013-0302.  Google Scholar

[42]

J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^{3}$, Calc. Var. Partial Differ. Equ., 48 (2013) 243-273. doi: 10.1007/s00526-012-0548-6.  Google Scholar

[43]

Z.-P. Wang and H.-S. Zhou, Positive solution for a nonlinear stationary Schr${{\rm{\ddot d}}}$inger-Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.  doi: 10.3934/dcds.2007.18.809.  Google Scholar

[44]

J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equation in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 37 (2010), 423-439.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

[45]

L. Xiao and J. Wang, Existence of positive solutions for a Schrödinger-Poisson system with critical growth, Appl. Anal., 99 (2020), 1827-1864.  doi: 10.1080/00036811.2018.1546004.  Google Scholar

[46]

L.-G. Zhao and F.-K. Zhao, On the existence of solutions for the Schrödinger-Poissom equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Comm. Contemp. Math., 10 (2008), 1-14.  doi: 10.1142/S021919970800282X.  Google Scholar

[2]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.  Google Scholar

[3]

A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[4]

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.  Google Scholar

[5]

A. AzzolliniP. d'venia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791.  doi: 10.1016/j.anihpc.2009.11.012.  Google Scholar

[6]

W. BaoN. J. Mauser and H.-P. Stimming, Effective one particle quantum dynamics of electrons: A numerical study of the Schrödinger-Poisson-X $\alpha$ model, Commun. Math. Sci., 1 (2003), 809-828.  doi: 10.4310/CMS.2003.v1.n4.a8.  Google Scholar

[7]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Poission equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019.  Google Scholar

[8]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168.  Google Scholar

[9]

V. Benci and D. Fortunato, Solitons in Schrödinger-Maxwell equations, J. Fixed Point Theory Appl., 15 (2014), 101-132.  doi: 10.1007/s11784-014-0184-1.  Google Scholar

[10]

H. Berestycki and P.L. Lions, Nonlinear scalar field equations II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.  Google Scholar

[11]

S. Bernstein, Sur une classe d'équations fonctionelles aux dérivées partielles, Bull. Acad. Sci. USSR Sér., 4 (1940), 17-26.,  Google Scholar

[12]

D. Bonheure and C. Mercuri, Embedding theorems and existence results for nonlinear Schrödinger-Poisson systems with unbounded and vanishing potentials, J. Diffierential Equations, 251 (2011), 1056-1085.  doi: 10.1016/j.jde.2011.04.010.  Google Scholar

[13]

I. CattoJ. DolbeaultO. Sanchez and J. Soler, Existence of steady states for the Maxwell-Schrödinger-Poisson system: Exploring the applicability of the concentration-compactness principle, Math. Models Methods Appl. Sci., 23 (2013), 1915-1938.  doi: 10.1142/S0218202513500541.  Google Scholar

[14]

G. CeramiR. Molle and D. Passaseo, Multiplicity of positive and nodal solutions for scalar field equations, J. Diff. Equations, 257 (2014), 3554-3606.  doi: 10.1016/j.jde.2014.07.002.  Google Scholar

[15]

G. Cerami and R. Molle, Positive bound state solution for some Schrödinger-Poisson systems, Nonlinearity, 29 (2016), 3013-3119.  doi: 10.1088/0951-7715/29/10/3103.  Google Scholar

[16]

G. Cerami and R. Molle, Infinitely many positive standing waves for Schrödinger equations with competing coefficients, Comm. Partial Differential Equations, 44 (2019), 73-109.  doi: 10.1080/03605302.2018.1541905.  Google Scholar

[17]

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Diffierential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017.  Google Scholar

[18]

G. CeramiD. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.  doi: 10.1002/cpa.21410.  Google Scholar

[19]

S.-M. ChangS. GustafsonK. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39 (2008), 1070-111.  doi: 10.1137/050648389.  Google Scholar

[20]

T. D'Aprile and J.-C. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differ. Equ., 25 (2006), 105-137.  doi: 10.1007/s00526-005-0342-9.  Google Scholar

[21]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon- Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.  doi: 10.1017/S030821050000353X.  Google Scholar

[22]

T. D'Aprile and D. Mugnai, Nonexistence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.  Google Scholar

[23]

M. Furtado, L. A. Maia and E. S. Medeiros, A note on the existence of a positive solution for a non-autonomous Schrödinger-Poisson system Analysis and Topology in Nonlinear Differential Equations, Progress in Nonlinear Differential Equations and their Applications, New York: Springer, 85 (2010), 277-286.  Google Scholar

[24]

B. GidasW.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[25]

X.-M. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 5 (2011), 869-889.  doi: 10.1007/s00033-011-0120-9.  Google Scholar

[26]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson system with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595.  doi: 10.1515/ans-2008-0305.  Google Scholar

[27]

Y.-S. Jiang and H.-S. Zhou, Bound states for a stationary nonlinear Schrödinger-Poisson system with signchanging potential in $\mathbb{R}^{3}$, Acta Mathematica Scientia, 29 (2009), 1095-1104.  doi: 10.1016/S0252-9602(09)60088-6.  Google Scholar

[28]

M.-K. Kwong, Uniqueness of positive solutions of $-\Delta u-u+u^p = 0$ in $\mathbb{R}^N$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[29]

E.-H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[30]

P.-L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, I-II, Ann. Inst. H. Poincaré Anal, Non Lineáire, 1 (1984), 109-145/223-283. doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[31]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.  doi: 10.1007/BF01205672.  Google Scholar

[32]

C. Mercuri, Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 Mat. Appl., 19 (2008), 211-227.  doi: 10.4171/RLM/520.  Google Scholar

[33]

C. Mercuri and T.-M. Tyler, On a class of nonlinear Schrödinger-Poisson systems involving a nonradial charge density, preprint, arXiv: 1805.00964. doi: 10.4171/rmi/1158.  Google Scholar

[34]

D. Mugnai, The Schrödinger-Poisson system with positive potential, Comm. Partial Diff. Eqns., 36 (2011), 1099-1117.  doi: 10.1080/03605302.2011.558551.  Google Scholar

[35]

E.-S. Noussair and J.-C. Wei, On the effect of the domain geometry on the existence and profile of nodal solution of some singularly perturbed semilinear Dirichlet problem, Indiana Univ. Math. J., 46 (1997), 1321-1332.  doi: 10.1512/iumj.1997.46.1401.  Google Scholar

[36]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Functional Analysis, 237 (2006), 655-674.  doi: 10.1016/j.jfa.2006.04.005.  Google Scholar

[37]

J. Slater, A simplification of the Hartree-Fock Method, Phys. Rev., 81 (1951), 385-390.   Google Scholar

[38]

J. SunT. Wu and Z. Feng, Non-autonomous Schrödinger-Poisson System in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933.  doi: 10.3934/dcds.2018077.  Google Scholar

[39]

G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297.  doi: 10.1007/s11587-011-0109-x.  Google Scholar

[40]

J. WangL. X. TianJ. X. Xu and F. B. 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.  Google Scholar

[41]

J. WangL. X. TianJ. X. Xu and F. B. Zhang, Existence and concentration of positive ground state solutions for semilinear Schrödinger-Poisson systems, Adv. Nonlinear Stud., 13 (2013), 553-582.  doi: 10.1515/ans-2013-0302.  Google Scholar

[42]

J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^{3}$, Calc. Var. Partial Differ. Equ., 48 (2013) 243-273. doi: 10.1007/s00526-012-0548-6.  Google Scholar

[43]

Z.-P. Wang and H.-S. Zhou, Positive solution for a nonlinear stationary Schr${{\rm{\ddot d}}}$inger-Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.  doi: 10.3934/dcds.2007.18.809.  Google Scholar

[44]

J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equation in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 37 (2010), 423-439.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

[45]

L. Xiao and J. Wang, Existence of positive solutions for a Schrödinger-Poisson system with critical growth, Appl. Anal., 99 (2020), 1827-1864.  doi: 10.1080/00036811.2018.1546004.  Google Scholar

[46]

L.-G. Zhao and F.-K. Zhao, On the existence of solutions for the Schrödinger-Poissom equations, J. Math. Anal. Appl., 346 (2008), 155-169.  doi: 10.1016/j.jmaa.2008.04.053.  Google Scholar

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