October  2021, 41(10): 4705-4736. 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  October 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (10) : 4705-4736. 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[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.

[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.

[37]

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

[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.

[39]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[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.

[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.

[37]

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

[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.

[39]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427

[2]

Lixi Wen, Wen Zhang. Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022109

[3]

Weiwei Ao, Juncheng Wei, Wen Yang. Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242

[4]

Lushun Wang, Minbo Yang, Yu Zheng. Infinitely many segregated solutions for coupled nonlinear Schrödinger systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6069-6102. doi: 10.3934/dcds.2019265

[5]

Dengfeng Lü. Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity. Communications on Pure and Applied Analysis, 2018, 17 (2) : 605-626. doi: 10.3934/cpaa.2018033

[6]

Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241

[7]

Caixia Chen, Aixia Qian. Multiple positive solutions for the Schrödinger-Poisson equation with critical growth. Mathematical Foundations of Computing, 2022, 5 (2) : 113-128. doi: 10.3934/mfc.2021036

[8]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094

[9]

Weiwei Ao, Liping Wang, Wei Yao. Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials. Communications on Pure and Applied Analysis, 2016, 15 (3) : 965-989. doi: 10.3934/cpaa.2016.15.965

[10]

Xinsheng Du, Qi Li, Zengqin Zhao, Gen Li. Bound state solutions for fractional Schrödinger-Poisson systems. Mathematical Foundations of Computing, 2022, 5 (1) : 57-66. doi: 10.3934/mfc.2021023

[11]

Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214

[12]

Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure and Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867

[13]

Rong Cheng, Jun Wang. Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021317

[14]

Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058

[15]

Xueqin Peng, Gao Jia. Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2325-2344. doi: 10.3934/dcdsb.2021134

[16]

Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071

[17]

Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104

[18]

Sitong Chen, Wennian Huang, Xianhua Tang. Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3055-3066. doi: 10.3934/dcdss.2020339

[19]

Marius Ghergu, Gurpreet Singh. On a class of mixed Choquard-Schrödinger-Poisson systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 297-309. doi: 10.3934/dcdss.2019021

[20]

Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (342)
  • HTML views (201)
  • Cited by (0)

Other articles
by authors

[Back to Top]