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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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