June  2018, 38(6): 3023-3032. doi: 10.3934/dcds.2018129

High energy solutions of the Choquard equation

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, Guangdong, China

2. 

RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100190, China

3. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author:Hang Li

Received  September 2017 Published  April 2018

In this paper we are concerned with the existence of positive high energy solutions of the Choquard equation. Under certain assumptions, the ground state of Choquard equation does not exist. However, by global compactness analysis, we prove that there exists a positive high energy solution.

Citation: Daomin Cao, Hang Li. High energy solutions of the Choquard equation. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3023-3032. doi: 10.3934/dcds.2018129
References:
[1]

A. Bahri and Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb{R}^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15. 

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Analysis, 99 (1987), 283-300.  doi: 10.1007/BF00282048.

[3]

D. Cao, Positive solution and bifurcation from the essential spactrum of a semilinear elliptic equation on $\mathbb{R}^N$, Nonlinear Analysis, 15 (1990), 1045-1052.  doi: 10.1016/0362-546X(90)90152-7.

[4]

D. Cao, Positive solution of a semilinear elliptic equation on $\mathbb{R}^N$, J.Partial Differential Equations, 8 (1995), 261-272. 

[5]

G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd ed, Cambridge University Press, 1952.

[6]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[7]

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

[8]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[9]

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

[10]

G. P. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Royal Soc. Edinburg Sect. A, 86 (1980), 291-301.  doi: 10.1017/S0308210500012191.

[11]

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.

[12]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Comm. Cont. Math. , 17 (2015), 1550005, 12 pp.

[13]

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

[14]

K. P. Tod and I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002.

[15]

G. Vaira, Existence of Bound States for Schrodinger-Newton Type Systems, Advanced Nonlinear Studies, 13 (2013), 495-516. 

[16]

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

[17]

T. Wang and T. Yi, Uniqueness of positive solutions of the Choquard type equations, Appl. Anal., 96 (2017), 409-417.  doi: 10.1080/00036811.2016.1138473.

[18]

C. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25 pp.

[19]

X. Zhu and D. Cao, The concentration-compact principle in nonlinear equations, Acta Math. Sci., 9 (1989), 307-328.  doi: 10.1016/S0252-9602(18)30356-4.

show all references

References:
[1]

A. Bahri and Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb{R}^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15. 

[2]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Analysis, 99 (1987), 283-300.  doi: 10.1007/BF00282048.

[3]

D. Cao, Positive solution and bifurcation from the essential spactrum of a semilinear elliptic equation on $\mathbb{R}^N$, Nonlinear Analysis, 15 (1990), 1045-1052.  doi: 10.1016/0362-546X(90)90152-7.

[4]

D. Cao, Positive solution of a semilinear elliptic equation on $\mathbb{R}^N$, J.Partial Differential Equations, 8 (1995), 261-272. 

[5]

G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd ed, Cambridge University Press, 1952.

[6]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[7]

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

[8]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.

[9]

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

[10]

G. P. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Royal Soc. Edinburg Sect. A, 86 (1980), 291-301.  doi: 10.1017/S0308210500012191.

[11]

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.

[12]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Comm. Cont. Math. , 17 (2015), 1550005, 12 pp.

[13]

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

[14]

K. P. Tod and I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002.

[15]

G. Vaira, Existence of Bound States for Schrodinger-Newton Type Systems, Advanced Nonlinear Studies, 13 (2013), 495-516. 

[16]

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

[17]

T. Wang and T. Yi, Uniqueness of positive solutions of the Choquard type equations, Appl. Anal., 96 (2017), 409-417.  doi: 10.1080/00036811.2016.1138473.

[18]

C. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25 pp.

[19]

X. Zhu and D. Cao, The concentration-compact principle in nonlinear equations, Acta Math. Sci., 9 (1989), 307-328.  doi: 10.1016/S0252-9602(18)30356-4.

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