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June  2021, 20(6): 2237-2256. doi: 10.3934/cpaa.2021065

Positive solutions for Choquard equation in exterior domains

1. 

School of mathematics and statistics, Hubei Normal University, Huangshi, 435002, China

2. 

School of mathematics and statistics, Wuhan University, Wuhan, 430072, China

* Corresponding author

Received  December 2020 Revised  February 2021 Published  June 2021 Early access  April 2021

Fund Project: P. Chen was supported by the Research Foundation of Education Bureau of Hubei Province, China(Grant No.Q20192505). X. Liu was supported by the NSFC (Grant No.11771342)

This work concerns with the following Choquard equation
$ \begin{equation*} \begin{cases} -\Delta u+ u = (\int_{\Omega}\frac{u^2(y)}{|x-y|^{N-2}}dy)u &{\rm{in }}\; \Omega ,\\ u\in H_0^1(\Omega), \end{cases} \end{equation*} $
where
$ \Omega\subseteq \mathbb{R}^{N} $
is an exterior domain with smooth boundary. We prove that the equation has at least one positive solution by variational and toplogical methods. Moreover, we establish a nonlocal version of global compactness result in unbounded domain.
Citation: Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2237-2256. doi: 10.3934/cpaa.2021065
References:
[1]

C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differ. Equ., 55 (2016), 48. doi: 10.1007/s00526-016-0984-9.

[2]

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

[3]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[4]

L. Battaglia and J. Van Schaftingen, Groundstates of the Choquard equations with a sign-changing self-interaction potential, Z. Angew. Math. Phys., 69 (2018), 16pp. doi: 10.1007/s00033-018-0975-0.

[5]

M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.  doi: 10.1016/j.jfa.2016.04.019.

[6]

M. Ghimenti and D. Pagliarrdini, Multiple positive solutions for a slightly subcritical Choquard problem on bounded domains, Calc. Var. Partial Differ. Equ., 58 (2019). doi: 10.1007/s00526-019-1605-1.

[7]

D. Goel and K. Sreenadh, Coron problem for nonlocal equations invloving Choquard nonlinearity, Adv. Nonlinear Stud., 20 (2020), 141-161.  doi: 10.1515/ans-2019-2064.

[8]

D. Goel and K. Sreenadh, Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains, Adv. Nonlinear Anal., 9 (2020), 803-835.  doi: 10.1515/anona-2020-0026.

[9]

F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.

[10]

F. GaoE D. da SilvaM. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration-compactness method, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 921-954.  doi: 10.1017/prm.2018.131.

[11]

L. Guo, T. Hu, S. Peng and W. Shuai, Existence and uniqueness of solutions for Choquard equation involving Hardy-Littlewood-Sobolev critical exponent, Calc. Var. Partial Differ. Equ., 58 (2019), 34 pp. doi: 10.1007/s00526-019-1585-1.

[12]

E.H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, second ed., 2001. doi: 10.1090/gsm/014.

[13]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/1977), 93-105.  doi: 10.1002/sapm197757293.

[14]

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

[15]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. 

[16]

X. Liu, S. Ma and X. Zhang, Infinitely many bound state solutions of Choquard equations with potentials, Z. Angew. Math. Phys., 69 (2018), 118. doi: 10.1007/s00033-018-1015-9.

[17]

V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.

[18]

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.

[19]

V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.

[20]

V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differ. Equ., 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.

[21]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12pp. doi: 10.1142/S0219199715500054.

[22]

L. Ma and Z. Lin, 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.

[23]

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.

[24]

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

[25]

J. Van Schaftingen and J. Xia, Choquard equations under confining external potentials, Nonlinear Differ. Equ. Appl., 24 (2017), 24pp. doi: 10.1007/s00030-016-0424-8.

[26]

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

[27]

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.

[28]

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

[29]

C. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differ. Equ., 55 (2016), 25pp. doi: 10.1007/s00526-016-1068-6.

[30]

J. Xia and Z. Wang, Saddle solutions for the Choquard equation, Calc. Var. Partial Differ. Equ., 58 (2019), 30pp. doi: 10.1007/s00526-019-1546-8.

show all references

References:
[1]

C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differ. Equ., 55 (2016), 48. doi: 10.1007/s00526-016-0984-9.

[2]

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

[3]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[4]

L. Battaglia and J. Van Schaftingen, Groundstates of the Choquard equations with a sign-changing self-interaction potential, Z. Angew. Math. Phys., 69 (2018), 16pp. doi: 10.1007/s00033-018-0975-0.

[5]

M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.  doi: 10.1016/j.jfa.2016.04.019.

[6]

M. Ghimenti and D. Pagliarrdini, Multiple positive solutions for a slightly subcritical Choquard problem on bounded domains, Calc. Var. Partial Differ. Equ., 58 (2019). doi: 10.1007/s00526-019-1605-1.

[7]

D. Goel and K. Sreenadh, Coron problem for nonlocal equations invloving Choquard nonlinearity, Adv. Nonlinear Stud., 20 (2020), 141-161.  doi: 10.1515/ans-2019-2064.

[8]

D. Goel and K. Sreenadh, Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains, Adv. Nonlinear Anal., 9 (2020), 803-835.  doi: 10.1515/anona-2020-0026.

[9]

F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.

[10]

F. GaoE D. da SilvaM. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration-compactness method, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 921-954.  doi: 10.1017/prm.2018.131.

[11]

L. Guo, T. Hu, S. Peng and W. Shuai, Existence and uniqueness of solutions for Choquard equation involving Hardy-Littlewood-Sobolev critical exponent, Calc. Var. Partial Differ. Equ., 58 (2019), 34 pp. doi: 10.1007/s00526-019-1585-1.

[12]

E.H. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, RI, second ed., 2001. doi: 10.1090/gsm/014.

[13]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/1977), 93-105.  doi: 10.1002/sapm197757293.

[14]

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

[15]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. 

[16]

X. Liu, S. Ma and X. Zhang, Infinitely many bound state solutions of Choquard equations with potentials, Z. Angew. Math. Phys., 69 (2018), 118. doi: 10.1007/s00033-018-1015-9.

[17]

V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.

[18]

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.

[19]

V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.

[20]

V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differ. Equ., 254 (2013), 3089-3145.  doi: 10.1016/j.jde.2012.12.019.

[21]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12pp. doi: 10.1142/S0219199715500054.

[22]

L. Ma and Z. Lin, 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.

[23]

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.

[24]

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

[25]

J. Van Schaftingen and J. Xia, Choquard equations under confining external potentials, Nonlinear Differ. Equ. Appl., 24 (2017), 24pp. doi: 10.1007/s00030-016-0424-8.

[26]

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

[27]

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.

[28]

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

[29]

C. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differ. Equ., 55 (2016), 25pp. doi: 10.1007/s00526-016-1068-6.

[30]

J. Xia and Z. Wang, Saddle solutions for the Choquard equation, Calc. Var. Partial Differ. Equ., 58 (2019), 30pp. doi: 10.1007/s00526-019-1546-8.

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