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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

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