GROUND STATE SOLUTIONS OF NONLINEAR SCHRÖDINGER EQUATIONS WITH ASYMPTOTICALLY PERIODIC POTENTIALS

Jianmin Guo; Shugui Kang; Shiwang Ma; Guang Zhang

doi:10.11948/20190137

2021 Volume 11 Issue 4

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Jianmin Guo, Shugui Kang, Shiwang Ma, Guang Zhang. GROUND STATE SOLUTIONS OF NONLINEAR SCHRÖDINGER EQUATIONS WITH ASYMPTOTICALLY PERIODIC POTENTIALS[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 1663-1677. doi: 10.11948/20190137

Citation:

Jianmin Guo, Shugui Kang, Shiwang Ma, Guang Zhang. GROUND STATE SOLUTIONS OF NONLINEAR SCHRÖDINGER EQUATIONS WITH ASYMPTOTICALLY PERIODIC POTENTIALS[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 1663-1677. doi: 10.11948/20190137

GROUND STATE SOLUTIONS OF NONLINEAR SCHRÖDINGER EQUATIONS WITH ASYMPTOTICALLY PERIODIC POTENTIALS

1.
School of Mathematics and Statistics, Shanxi Datong University, Xingyun Street, 037009, China
2.
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
3.
School of Science, Tianjin University of Commerce, Tianjin 300134, China

Corresponding author: Email address: shiwangm@nankai.edu.cn (S. Ma)
Fund Project: The authors were supported by National Natural Science Foundation of China (Grant Nos. 11871314, 11571187, 11771182), National Natural Science Foundation of Shanxi Province (201801D121001), Applied Foundational Research Project of the Datong Science and Technology Bureau(2019154), Science and Technology Planning Project of Pingcheng District of Datong City (201906) and Scientific Research Project of Shanxi Datong University

Abstract

Abstract

In this paper, by using a concentration-compactness argument, we study the existence of ground state solutions for nonlinear Schrödinger equations with asymptotically periodic potentials. In particular, when the coefficients are "competing", some sufficient conditions are given to guarantee the existence of ground state solutions, which improve and generalize some previous results in the literature.
- Nonlinear Schrödinger equation /
- asymptotically periodic /
- ground state solution
MSC: 35J10, 35J20, 35J61

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References

[1]	C. O. Alves and G. F. Germano, Ground state solution for a class of indefinite variational problems with critical growth, J. Diff. Eqs, 2018, 265, 444-477. doi: 10.1016/j.jde.2018.02.039 CrossRef Google Scholar
[2]	A. Bahri and Y. Li. On a min-max procedure for the existence of a positive solution for certain scalar field equations in RN, Rev. Mat. Iber, 1990, 6, 1-15. Google Scholar
[3]	A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. Henri Poincaré Analyse non linéaire, 1997, 14(3), 365-413. doi: 10.1016/S0294-1449(97)80142-4 CrossRef Google Scholar
[4]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Archive for Rational Mechanics and Analysis, 1983, 82(4), 313-345. doi: 10.1007/BF00250555 CrossRef Google Scholar
[5]	G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan Journal of Mathematics, 2006, 74(1), 47-77. doi: 10.1007/s00032-006-0059-z CrossRef Google Scholar
[6]	G. Cerami and A. Pomponio, On Some Scalar Field Equations with Competing Coefficients, Mathematics, 2015, 1-19. Google Scholar
[7]	G. Cerami and A. Pomponio, On Some Scalar Field Equations with Competing Coefficients, International Mathematics Research Notices, 2018, 8, 2481-2507. Google Scholar
[8]	W. Ding and W. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal, 1986, 91, 283-308. doi: 10.1007/BF00282336 CrossRef Google Scholar
[9]	G. Evéquoz and T. Weth, Entire Solutions to Nonlinear Scalar Field Equations with Indefinite Linear Part, Advanced Nonlinear Studies, 2012, 12, 281-314. doi: 10.1515/ans-2012-0206 CrossRef Google Scholar
[10]	T. Hu and L. Lu, On some nonlocal equations with competing coefficients, J. Math. Anal. Appl., 2018, 460, 863-884. doi: 10.1016/j.jmaa.2017.12.027 CrossRef Google Scholar
[11]	X. Lin and X. Tang, Nehari-Type Ground State Positive Solutions for Superlinear Asymptotically Periodic Schrödinger Equations, Abstract and Applied Analysis, 2014, 2014, 1-7. Google Scholar
[12]	H. Lins and E. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Analysis, 2009, 71(7), 2890-2905. doi: 10.1007/s00526-009-0299-1 CrossRef Google Scholar
[13]	P. H. Lions, The concentration-compactness principle in the calculus of variations: The locally compact cases, Part Ⅰ and Part Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1984, 1, 223-283. doi: 10.1016/S0294-1449(16)30422-X CrossRef Google Scholar
[14]	X. Liu, J. Liu and Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var., 2013, 46, 641-669. doi: 10.1007/s00526-012-0497-0 CrossRef Google Scholar
[15]	R. De Marchi, Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh, 2015, 145(04), 745-757. doi: 10.1017/S0308210515000104 CrossRef Google Scholar
[16]	V. Moroz and J. Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differential Equations, 2013, 254(8), 3089-3145. doi: 10.1016/j.jde.2012.12.019 CrossRef Google Scholar
[17]	V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotices, J. Functional Analysis, 2013, 265(2), 153-184. doi: 10.1016/j.jfa.2013.04.007 CrossRef Google Scholar
[18]	X. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 2015, 58, 715-728. doi: 10.1007/s11425-014-4957-1 CrossRef Google Scholar
[19]	J. Wang, M. Qu and L. Xiao, Existence of positive solutions to the nonlinear Choquard equation with competing potentials, Electronic J. Differential Equations, 2018, 2018, 1-21. Google Scholar
[20]	M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar
[21]	Y. Xue, J. Liu and C. Tang, A ground state solution for an asymptotically periodic quasilinear Schrödinger equation, Comput. Math. Appl, 2017, 74(6), 1143-1157. doi: 10.1016/j.camwa.2017.05.033 CrossRef Google Scholar
[22]	Y. Xue and C. Tang, Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth, Comm. Pur. Appl. Anal., 2018, 17(3), 1121-1145. doi: 10.3934/cpaa.2018054 CrossRef Google Scholar