American Institute of Mathematical Sciences

Advanced
September  2017, 37(9): 4973-5002. doi: 10.3934/dcds.2017214

Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials

Xianhua Tang , and  Sitong Chen

School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China

* Corresponding author

Received  February 2017 Revised  April 2017 Published  June 2017

Fund Project: This work is partially supported by the National Natural Science Foundation of China (No: 11571370).

This paper is dedicated to studying the following Schrödinger-Poisson problem
$\left\{ \begin{array}{ll} -\triangle u+V(x)u+φ u=f(u), \ \ \ \ x∈ \mathbb{R}^{3},\\ -\triangle φ=u^2, \ \ \ \ x∈ \mathbb{R}^{3}, \end{array} \right. $
where
$V(x)$
 is weakly differentiable and
$f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$
. By introducing some new tricks, we prove the above problem admits a ground state solution of Nehari-Pohozaev type and a least energy solution under mild assumptions on
$V$
 and
$f$
. Our results generalize and improve the ones in [D. Ruiz, J. Funct. Anal. 237 (2006) 655-674], [J.J. Sun, S.W. Ma, J. Differential Equations 260 (2016) 2119-2149] and some related literature.
Keywords: Schrödinger-Poisson problem, ground state solution of NehariPohozaev type, the least energy solutions.
Mathematics Subject Classification: 35J10, 35J20.
Citation: Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214
References:
[1]

N. Ackermann and T. Weth, Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math., 7 (2005), 269-298.  doi: 10.1142/S0219199705001763.  Google Scholar

[2]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

R. Benguria, H. Brezis and E. H. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.  doi: 10.1007/BF01942059.  Google Scholar

[7]

I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.  doi: 10.1080/03605309208820878.  Google Scholar

[8]

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.  Google Scholar

[9]

S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$ Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2.  Google Scholar

[10]

S. T. Chen and X. H. Tang, Nehari type ground state solutions for asymptotically periodic Schrödinger-Poisson systems, Taiwan. J. Math., 21 (2017), 363-383.  doi: 10.11650/tjm/7784.  Google Scholar

[11]

G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.   Google Scholar

[12]

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

[13]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.   Google Scholar

[14] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific, Singapore, 2007.  doi: 10.1142/9789812709639.  Google Scholar
[15]

X. M. 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.  Google Scholar

[16]

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

[17]

L. R. Huang, E. M. Rocha and J. Q. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), 2463-2483.  doi: 10.1016/j.jde.2013.06.022.  Google Scholar

[18]

W. N. Huang and X. H. Tang, Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 415 (2014), 791-802.  doi: 10.1016/j.jmaa.2014.02.015.  Google Scholar

[19]

W. N. Huang and X. H. Tang, Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations with critical nonlinearity, Taiwan. J. Math., 18 (2014), 1203-1217.  doi: 10.11650/tjm.18.2014.3993.  Google Scholar

[20]

L. Jeanjean, On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. -A, 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar

[21]

L. Jeanjean and J. Toland, Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. Ⅰ Math., 327 (1998), 23-28.  doi: 10.1016/S0764-4442(98)80097-9.  Google Scholar

[22]

E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.  doi: 10.1103/RevModPhys.53.603.  Google Scholar

[23]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[24]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.  Google Scholar

[25]

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

[26] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.  doi: 10.1007/978-3-7091-6961-2.  Google Scholar
[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

J. Seok, On nonlinear Schrödinger-Poisson equations with general potentials, J. Math. Anal. Appl., 401 (2013), 672-681.  doi: 10.1016/j.jmaa.2012.12.054.  Google Scholar

[30]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar

[31]

J. J. Sun and S. W. 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.  Google Scholar

[32]

X. H. Tang and B. T. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.  doi: 10.1016/j.jde.2016.04.032.  Google Scholar

[33] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.  doi: 10.1007/978-1-4612-4146-1.  Google Scholar
[34]

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

[35]

L. G. Zhao and F. K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.  doi: 10.1016/j.na.2008.02.116.  Google Scholar

show all references

References:
[1]

N. Ackermann and T. Weth, Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math., 7 (2005), 269-298.  doi: 10.1142/S0219199705001763.  Google Scholar

[2]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

R. Benguria, H. Brezis and E. H. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.  doi: 10.1007/BF01942059.  Google Scholar

[7]

I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.  doi: 10.1080/03605309208820878.  Google Scholar

[8]

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.  Google Scholar

[9]

S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$ Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2.  Google Scholar

[10]

S. T. Chen and X. H. Tang, Nehari type ground state solutions for asymptotically periodic Schrödinger-Poisson systems, Taiwan. J. Math., 21 (2017), 363-383.  doi: 10.11650/tjm/7784.  Google Scholar

[11]

G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.   Google Scholar

[12]

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

[13]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.   Google Scholar

[14] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific, Singapore, 2007.  doi: 10.1142/9789812709639.  Google Scholar
[15]

X. M. 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.  Google Scholar

[16]

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

[17]

L. R. Huang, E. M. Rocha and J. Q. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), 2463-2483.  doi: 10.1016/j.jde.2013.06.022.  Google Scholar

[18]

W. N. Huang and X. H. Tang, Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 415 (2014), 791-802.  doi: 10.1016/j.jmaa.2014.02.015.  Google Scholar

[19]

W. N. Huang and X. H. Tang, Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations with critical nonlinearity, Taiwan. J. Math., 18 (2014), 1203-1217.  doi: 10.11650/tjm.18.2014.3993.  Google Scholar

[20]

L. Jeanjean, On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. -A, 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar

[21]

L. Jeanjean and J. Toland, Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. Ⅰ Math., 327 (1998), 23-28.  doi: 10.1016/S0764-4442(98)80097-9.  Google Scholar

[22]

E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.  doi: 10.1103/RevModPhys.53.603.  Google Scholar

[23]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. of Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[24]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.  Google Scholar

[25]

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

[26] P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.  doi: 10.1007/978-3-7091-6961-2.  Google Scholar
[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

J. Seok, On nonlinear Schrödinger-Poisson equations with general potentials, J. Math. Anal. Appl., 401 (2013), 672-681.  doi: 10.1016/j.jmaa.2012.12.054.  Google Scholar

[30]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.  Google Scholar

[31]

J. J. Sun and S. W. 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.  Google Scholar

[32]

X. H. Tang and B. T. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.  doi: 10.1016/j.jde.2016.04.032.  Google Scholar

[33] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.  doi: 10.1007/978-1-4612-4146-1.  Google Scholar
[34]

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

[35]

L. G. Zhao and F. K. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.  doi: 10.1016/j.na.2008.02.116.  Google Scholar

[1]

Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257
[2]

Xiaoping Chen, Chunlei Tang. Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2291-2312. doi: 10.3934/cpaa.2021077
[3]

Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2805-2816. doi: 10.3934/jimo.2020095
[4]

Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
[5]

Sitong Chen, Wennian Huang, Xianhua Tang. Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3055-3066. doi: 10.3934/dcdss.2020339
[6]

Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079
[7]

Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[8]

Xinsheng Du, Qi Li, Zengqin Zhao, Gen Li. Bound state solutions for fractional Schrödinger-Poisson systems. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021023
[9]

Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064
[10]

Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure & Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867
[11]

Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
[12]

Yi He, Lu Lu, Wei Shuai. Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103
[13]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, 2021, 29 (3) : 2475-2488. doi: 10.3934/era.2020125
[14]

Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3331-3355. doi: 10.3934/cpaa.2021108
[15]

Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
[16]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99
[17]

Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048
[18]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[19]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[20]

Zhengping Wang, Huan-Song Zhou. Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 809-816. doi: 10.3934/dcds.2007.18.809

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (558)
  • HTML views (143)
  • Cited by (63)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences