April  2019, 12(2): 297-309. doi: 10.3934/dcdss.2019021

On a class of mixed Choquard-Schrödinger-Poisson systems

School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

* Corresponding author: Marius Ghergu

To Professor Vicenţiu Rǎdulescu on the occasion of his 60th anniversary with deep esteem and consideration

Received  May 2017 Revised  November 2017 Published  August 2018

We study the system
$\left\{\begin{split}-\Delta u+u&=(I_\alpha*|u|^p)|u|^{p-2}u+K(x) \phi |u|^{q-2}u & \qquad \mbox{ in }\mathbb{R}^N,\\-\Delta \phi&=K(x)|u|^q& \qquad \mbox{ in }\mathbb{R}^N,\end{split}\right.$
where
$N≥ 3$
,
$α∈ (0,N)$
,
$p,q>1$
and
$K≥ 0$
. Using a Pohozaev type identity we first derive conditions in terms of
$p,q,N,α$
and
$K$
for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach.
Citation: Marius Ghergu, Gurpreet Singh. On a class of mixed Choquard-Schrödinger-Poisson systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 297-309. doi: 10.3934/dcdss.2019021
References:
[1]

C. O. AlvesG. M. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Advances in Nonlinear Analysis, 5 (2016), 331-345.  doi: 10.1515/anona-2015-0123.  Google Scholar

[2]

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

[3]

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

[4]

V. Bogachev, Measure Theory, Springer, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[5]

O. BokanowskiJ. L. López and J. Soler, On an exchange interaction model for quantum transport: The Schrödinger-Poisson-Slater system, Math. Models Methods Appl. Sci., 13 (2003), 1397-1412.  doi: 10.1142/S0218202503002969.  Google Scholar

[6]

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[7]

I. CattoJ. DolbeaultO. Sánchez 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

[8]

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

[9]

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

[10]

Y.-H. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.  doi: 10.1088/0951-7715/29/6/1827.  Google Scholar

[11]

P. D'AveniaG. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.  Google Scholar

[12]

J. T. Devreese and A. S. Alexandrov, Advances in Polaron Physics, Springer Series in SolidState Sciences, Springer, 159 (2010). Google Scholar

[13]

M. Ghergu and S. D. Taliaferro, Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity, J. Differential Equations, 261 (2016), 189-217.  doi: 10.1016/j.jde.2016.03.004.  Google Scholar

[14]

K. R. W. Jones, Newtonian quantum gravity, Australian J. Phys., 48 (1995), 1055-1081.  doi: 10.1071/PH951055.  Google Scholar

[15]

C. Le Bris and P.-L. Lions, From atoms to crystals: A mathematical journey, Bull. Amer. Math. Soc. (N.S.), 42 (2005), 291-363.  doi: 10.1090/S0273-0979-05-01059-1.  Google Scholar

[16]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies Appl. Math., 57 (1976/77), 93-105.   Google Scholar

[17]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[18]

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

[19]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[20]

N. J. Mauser, The Schrödinger-Poisson-$X_α$ equation, Appl. Math. Lett., 14 (2001), 759-763.  doi: 10.1016/S0893-9659(01)80038-0.  Google Scholar

[21]

C. Mercuri, V. Moroz and J. Van Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at critical frequency, Calculus of Variations and Partial Differential Equations, 55 (2016), p146, arXiv: 1507.02837 doi: 10.1007/s00526-016-1079-3.  Google Scholar

[22]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[23]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[24]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar

[25]

R. Penrose, On gravity's role in quantum state reduction, General Relativity and Gravitation, 28 (1996), 581-600.  doi: 10.1007/BF02105068.  Google Scholar

[26]

G. Singh, Nonlocal perturbations of the fractional Choquard equation, Adv. Nonlinear Anal., (2017), in press. doi: 10.1515/anona-2017-0126.  Google Scholar

[27]

J. Slater, A Simplification of the Hartree-Fock Method, Phys. Rev., 81 (1951), 385-390.  doi: 10.1016/B978-0-08-017819-6.50031-9.  Google Scholar

[28]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22pp. doi: 10.1063/1.3060169.  Google Scholar

[29]

M. Willem, Functional Analysis: Fundamentals and Applications, Cornerstones, vol. XIV, Birkhäuser, Basel, 2013. doi: 10.1007/978-1-4614-7004-5.  Google Scholar

show all references

References:
[1]

C. O. AlvesG. M. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Advances in Nonlinear Analysis, 5 (2016), 331-345.  doi: 10.1515/anona-2015-0123.  Google Scholar

[2]

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

[3]

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

[4]

V. Bogachev, Measure Theory, Springer, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[5]

O. BokanowskiJ. L. López and J. Soler, On an exchange interaction model for quantum transport: The Schrödinger-Poisson-Slater system, Math. Models Methods Appl. Sci., 13 (2003), 1397-1412.  doi: 10.1142/S0218202503002969.  Google Scholar

[6]

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[7]

I. CattoJ. DolbeaultO. Sánchez 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

[8]

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

[9]

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

[10]

Y.-H. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.  doi: 10.1088/0951-7715/29/6/1827.  Google Scholar

[11]

P. D'AveniaG. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.  Google Scholar

[12]

J. T. Devreese and A. S. Alexandrov, Advances in Polaron Physics, Springer Series in SolidState Sciences, Springer, 159 (2010). Google Scholar

[13]

M. Ghergu and S. D. Taliaferro, Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity, J. Differential Equations, 261 (2016), 189-217.  doi: 10.1016/j.jde.2016.03.004.  Google Scholar

[14]

K. R. W. Jones, Newtonian quantum gravity, Australian J. Phys., 48 (1995), 1055-1081.  doi: 10.1071/PH951055.  Google Scholar

[15]

C. Le Bris and P.-L. Lions, From atoms to crystals: A mathematical journey, Bull. Amer. Math. Soc. (N.S.), 42 (2005), 291-363.  doi: 10.1090/S0273-0979-05-01059-1.  Google Scholar

[16]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies Appl. Math., 57 (1976/77), 93-105.   Google Scholar

[17]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[18]

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

[19]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar

[20]

N. J. Mauser, The Schrödinger-Poisson-$X_α$ equation, Appl. Math. Lett., 14 (2001), 759-763.  doi: 10.1016/S0893-9659(01)80038-0.  Google Scholar

[21]

C. Mercuri, V. Moroz and J. Van Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at critical frequency, Calculus of Variations and Partial Differential Equations, 55 (2016), p146, arXiv: 1507.02837 doi: 10.1007/s00526-016-1079-3.  Google Scholar

[22]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.  Google Scholar

[23]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[24]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar

[25]

R. Penrose, On gravity's role in quantum state reduction, General Relativity and Gravitation, 28 (1996), 581-600.  doi: 10.1007/BF02105068.  Google Scholar

[26]

G. Singh, Nonlocal perturbations of the fractional Choquard equation, Adv. Nonlinear Anal., (2017), in press. doi: 10.1515/anona-2017-0126.  Google Scholar

[27]

J. Slater, A Simplification of the Hartree-Fock Method, Phys. Rev., 81 (1951), 385-390.  doi: 10.1016/B978-0-08-017819-6.50031-9.  Google Scholar

[28]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22pp. doi: 10.1063/1.3060169.  Google Scholar

[29]

M. Willem, Functional Analysis: Fundamentals and Applications, Cornerstones, vol. XIV, Birkhäuser, Basel, 2013. doi: 10.1007/978-1-4614-7004-5.  Google Scholar

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