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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 and 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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[25]

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

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

[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.

[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.

[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.

[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.

[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.

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

[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.

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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[25]

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

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

[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.

[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.

[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.

[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.

[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.

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

[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.

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