# American Institute of Mathematical Sciences

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 and Applied Analysis, 2020, 19 (3) : 1351-1365. doi: 10.3934/cpaa.2020066
##### References:
 [1] C. O. Alves, G. 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. [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. [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. [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. [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. [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. [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. [8] J. M. do Ó, E. 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. [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. [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. [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. [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. [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. [14] J.-Q. Liu, Y.-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. [15] J. Liu, J.-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. [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. [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. [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. [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. [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. [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. [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. [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. [24] S. I. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie-verlag, 1954. [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. [26] Z. Shen, F. Gao and M. Yang, On critical Choquard equation with potential well, Discrete Contin. Dyn. Syst., 38 (2018), 3567-3593.  doi: 10.3934/dcds.2018151. [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. [28] M. Willem, Minimax Theorems, vol. 24, Springer Science and Business Media, 1997. doi: 10.1007/978-1-4612-4146-1. [29] H. Zhang, J. 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. [30] H. Zhang, J. 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.

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##### References:
 [1] C. O. Alves, G. 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. [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. [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. [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. [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. [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. [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. [8] J. M. do Ó, E. 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. [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. [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. [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. [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. [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. [14] J.-Q. Liu, Y.-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. [15] J. Liu, J.-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. [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. [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. [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. [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. [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. [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. [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. [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. [24] S. I. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie-verlag, 1954. [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. [26] Z. Shen, F. Gao and M. Yang, On critical Choquard equation with potential well, Discrete Contin. Dyn. Syst., 38 (2018), 3567-3593.  doi: 10.3934/dcds.2018151. [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. [28] M. Willem, Minimax Theorems, vol. 24, Springer Science and Business Media, 1997. doi: 10.1007/978-1-4612-4146-1. [29] H. Zhang, J. 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. [30] H. Zhang, J. 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.
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