March  2020, 19(3): 1351-1365. doi: 10.3934/cpaa.2020066

A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author

Received  March 2019 Revised  August 2019 Published  November 2019

Fund Project: This work is partially supported by National Natural Science Foundation of China (No.11971393) and Graduate Student Scientific Research Innovation Projects in Chongqing (No. CYB19082).

In this paper, we investigate the following Choquard equation
$ \begin{equation*} -\Delta u+V(x)u = \lambda(I_\alpha*F(u))f(u) \ \ \ \ \ \ {\rm in} \ \mathbb{R}^N, \end{equation*} $
where
$ N\geq 3, \lambda>0, \alpha\in (0, N) $
,
$ V $
is an asymptotically periodic potential,
$ I_\alpha $
is the Riesz potential, the nonlinearity term
$ F(s) = \int_{0}^{s}f(t)dt $
and
$ f $
is only locally defined in a neighborhood of
$ u = 0 $
and satisfies the suitable conditions. By using the Nehari manifold and the Moser iteration, we prove the existence of positive solutions for the equation with sufficiently large
$ \lambda $
.
Citation: Gui-Dong Li, Yong-Yong Li, Xiao-Qi Liu, Chun-Lei Tang. A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1351-1365. doi: 10.3934/cpaa.2020066
References:
[1]

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

[2]

C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55 (2016), Art. 48, 28. doi: 10.1007/s00526-016-0984-9.  Google Scholar

[3]

C. O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations, 257 (2014), 4133-4164.  doi: 10.1016/j.jde.2014.08.004.  Google Scholar

[4]

T. Bartsch, Z.-Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, in Stationary Partial Differential Equations. Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, 1–55. doi: 10.1016/S1874-5733(05)80009-9.  Google Scholar

[5]

S. Chen and L. Xiao, Existence of a nontrivial solution for a strongly indefinite periodic Choquard system, Calc. Var. Partial Differential Equations, 54 (2015), 599-614.  doi: 10.1007/s00526-014-0797-7.  Google Scholar

[6]

C. Chu and H. Liu, Existence of positive solutions for a quasilinear Schrödinger equation, Nonlinear Anal. Real World Appl., 44 (2018), 118-127.  doi: 10.1016/j.nonrwa.2018.04.007.  Google Scholar

[7]

D. G. Costa and Z.-Q. Wang, Multiplicity results for a class of superlinear elliptic problems, Proc. Amer. Math. Soc., 133 (2005), 787-794.  doi: 10.1090/S0002-9939-04-07635-X.  Google Scholar

[8]

J. M. do ÓE. Medeiros and U. Severo, On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations, J. Math. Anal. Appl., 342 (2008), 432-445.  doi: 10.1016/j.jmaa.2007.11.058.  Google Scholar

[9]

F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., 20 (2018), 1750037, 22. doi: 10.1142/S0219199717500377.  Google Scholar

[10]

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

[11]

L. Li and X. Zhong, Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities, J. Math. Anal. Appl., 435 (2016), 955-967.  doi: 10.1016/j.jmaa.2015.10.075.  Google Scholar

[12]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar

[13]

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

[14]

J.-Q. LiuY.-Q. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[15]

J. LiuJ.-F. Liao and C.-L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl., 71 (2016), 965-976.  doi: 10.1016/j.camwa.2016.01.004.  Google Scholar

[16]

S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016.  Google Scholar

[17]

X. Liu, S. Ma and X. Zhang, Infinitely many bound state solutions of Choquard equations with potentials, Z. Angew. Math. Phys., 69 (2018), Art. 118, 29. doi: 10.1007/s00033-018-1015-9.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.  doi: 10.1007/s00526-014-0709-x.  Google Scholar

[23]

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

[24]

S. I. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie-verlag, 1954. Google Scholar

[25]

M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc. (2), 44 (1991), 491–502. doi: 10.1112/jlms/s2-44.3.491.  Google Scholar

[26]

Z. ShenF. Gao and M. Yang, On critical Choquard equation with potential well, Discrete Contin. Dyn. Syst., 38 (2018), 3567-3593.  doi: 10.3934/dcds.2018151.  Google Scholar

[27]

J. Van Schaftingen and J. Xia, Choquard equations under confining external potentials, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 1, 24. doi: 10.1007/s00030-016-0424-8.  Google Scholar

[28]

M. Willem, Minimax Theorems, vol. 24, Springer Science and Business Media, 1997. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[29]

H. ZhangJ. Xu and F. Zhang, Bound and ground states for a concave-convex generalized Choquard equation, Acta Appl. Math., 147 (2017), 81-93.  doi: 10.1007/s10440-016-0069-y.  Google Scholar

[30]

H. ZhangJ. Xu and F. Zhang, Existence and multiplicity of solutions for a generalized Choquard equation, Comput. Math. Appl., 73 (2017), 1803-1814.  doi: 10.1016/j.camwa.2017.02.026.  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, Adv. Nonlinear Anal., 5 (2016), 331-345.  doi: 10.1515/anona-2015-0123.  Google Scholar

[2]

C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55 (2016), Art. 48, 28. doi: 10.1007/s00526-016-0984-9.  Google Scholar

[3]

C. O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations, 257 (2014), 4133-4164.  doi: 10.1016/j.jde.2014.08.004.  Google Scholar

[4]

T. Bartsch, Z.-Q. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations, in Stationary Partial Differential Equations. Vol. II, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, 1–55. doi: 10.1016/S1874-5733(05)80009-9.  Google Scholar

[5]

S. Chen and L. Xiao, Existence of a nontrivial solution for a strongly indefinite periodic Choquard system, Calc. Var. Partial Differential Equations, 54 (2015), 599-614.  doi: 10.1007/s00526-014-0797-7.  Google Scholar

[6]

C. Chu and H. Liu, Existence of positive solutions for a quasilinear Schrödinger equation, Nonlinear Anal. Real World Appl., 44 (2018), 118-127.  doi: 10.1016/j.nonrwa.2018.04.007.  Google Scholar

[7]

D. G. Costa and Z.-Q. Wang, Multiplicity results for a class of superlinear elliptic problems, Proc. Amer. Math. Soc., 133 (2005), 787-794.  doi: 10.1090/S0002-9939-04-07635-X.  Google Scholar

[8]

J. M. do ÓE. Medeiros and U. Severo, On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations, J. Math. Anal. Appl., 342 (2008), 432-445.  doi: 10.1016/j.jmaa.2007.11.058.  Google Scholar

[9]

F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., 20 (2018), 1750037, 22. doi: 10.1142/S0219199717500377.  Google Scholar

[10]

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

[11]

L. Li and X. Zhong, Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities, J. Math. Anal. Appl., 435 (2016), 955-967.  doi: 10.1016/j.jmaa.2015.10.075.  Google Scholar

[12]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.  Google Scholar

[13]

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

[14]

J.-Q. LiuY.-Q. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[15]

J. LiuJ.-F. Liao and C.-L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl., 71 (2016), 965-976.  doi: 10.1016/j.camwa.2016.01.004.  Google Scholar

[16]

S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016.  Google Scholar

[17]

X. Liu, S. Ma and X. Zhang, Infinitely many bound state solutions of Choquard equations with potentials, Z. Angew. Math. Phys., 69 (2018), Art. 118, 29. doi: 10.1007/s00033-018-1015-9.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.  doi: 10.1007/s00526-014-0709-x.  Google Scholar

[23]

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

[24]

S. I. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie-verlag, 1954. Google Scholar

[25]

M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc. (2), 44 (1991), 491–502. doi: 10.1112/jlms/s2-44.3.491.  Google Scholar

[26]

Z. ShenF. Gao and M. Yang, On critical Choquard equation with potential well, Discrete Contin. Dyn. Syst., 38 (2018), 3567-3593.  doi: 10.3934/dcds.2018151.  Google Scholar

[27]

J. Van Schaftingen and J. Xia, Choquard equations under confining external potentials, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 1, 24. doi: 10.1007/s00030-016-0424-8.  Google Scholar

[28]

M. Willem, Minimax Theorems, vol. 24, Springer Science and Business Media, 1997. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[29]

H. ZhangJ. Xu and F. Zhang, Bound and ground states for a concave-convex generalized Choquard equation, Acta Appl. Math., 147 (2017), 81-93.  doi: 10.1007/s10440-016-0069-y.  Google Scholar

[30]

H. ZhangJ. Xu and F. Zhang, Existence and multiplicity of solutions for a generalized Choquard equation, Comput. Math. Appl., 73 (2017), 1803-1814.  doi: 10.1016/j.camwa.2017.02.026.  Google Scholar

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