Abstract
We study the existence of ground state solutions for a class of discrete nonlinear Schrödinger equation with a sign-changing potential which is periodic or asymptotically periodic. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the ℤ-translation invariance, and many effective methods for periodic problems cannot be applied to asymptotically periodic ones. These enables us to develop a direct approach to find ground state solutions with asymptotically periodic potential. Two types of ground state solutions are obtained with some new super-quadratic conditions on nonlinearity which are weaker that some well-known ones. Moreover, our conditions can also be used to significantly improve the well-known results of the corresponding continuous nonlinear Schrödinger equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brazhnyi V.A., Konotop V.V., Theory of nonlinear matter waves in optical lattices. Modern Phys. Lett. B, 2004, 18(14): 627–651
Chen J.H., Cheng B.T., Huang X.J., Ground state solutions for a class of quasilinear Schrödinger equations with Choquard type nonlinearity. Appl. Math. Lett., 2020, 102: 106141, 7 pp.
Chen G.W., Ma S.W., Discrete nonlinear Schrödinger equations with superlinear nonlinearities. Appl. Math. Comput., 2012, 218(9): 5496–5507
Chen G.W., Ma S.W., Ground state and geometrically distinct solitons of discrete nonlinear Schrödinger equations with saturable nonlinearities. Stud. Appl. Math., 2013, 131(4): 389–413
Chen G.W., Ma S.W., Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or superlinear terms. Appl. Math. Comput., 2014, 232: 787–798
Chen G.W., Schechter M., Non-periodic discrete Schrödinger equations: ground state solutions. Z. Angew. Math. Phys., 2016, 67 (3): Art. 72, 15 pp.
Christiansen P.L., Scott A.C., Davydov’s Soliton Revisited: Self-trapping of Vibrational Energy in Protein. New York, NY: Springer, 1990
Christodoulides D.N., Lederer F., Silberberg Y., Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature, 2003, 424: 817–823
Davydov A.S., Solitons and energy transfer along protein molecules. J. Theoret. Biol., 1977, 66(2): 379–387
Flach S., Gorbach A.V., Discrete breathers—advance in theory and applications. Phys. Rep., 2008, 467(1–3): 1–116
Kevrekidis P.G., Rasmussen K.Ø., Bishop A.R., The discrete nonlinear Schrödinger equation: a survey of recent results. Internat. J. Modern Phys. B, 2001, 15(21): 2833–2900
Kivshar Y.S., Agrawal G.P., Optical Solitons: From Fibers to Photonic Crystals. San Diego, CA: Academic Press, 2003
Kopidakis G., Aubry S., Tsironis G.P., Targeted energy transfer through discrete breathers in nonlinear systems. Phys. Rev. Lett., 2001, 87: 165501, 4 pp.
Li G.B., Szulkin A., An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math., 2002, 4(4): 763–776
Lin G.H., Zhou Z., Yu J.S., Ground state solutions of discrete asymptotically linear Schrödinger equations with bounded and non-periodic potentials. J. Dynam. Differential Equations, 2020, 32(2): 527–555
Livi R., Franzosi R., Oppo G.-L., Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation. Phys. Rev. Lett., 2006, 97: 060401
Mawhin J., Periodic solutions of second order nonlinear difference systems with ϕ-Laplacian: a variational approach. Nonlinear Anal., 2012, 75(12): 4672–4687
Morsch O., Oberthaler M., Dynamics of Bose–Einstein condensates in optical lattices. Rev. Mod. Phys., 2006, 78: 179–215
Pankov A., Gap solitons in periodic discrete nonlinear Schrödinger equations. Nonlinearity, 2006, 19(1): 27–40
Pankov A., Gap solitons in periodic discrete nonlinear Schrödinger equations, II. A generalized Nehari manifold approach. Discrete Contin. Dyn. Syst., 2007, 19(2): 419–430
Pankov A., Rothos V., Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2008, 464(2100): 3219–3236
Pankov A., Zhang G., Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearity. J. Math. Sci. (N.Y.), 2011, 177(1): 71–82
Schechter M., Zou W.M., Weak linking theorems and Schrödinger equations with critical Sobolev exponent. ESAIM Control Optim. Calc. Var., 2003, 9: 601–619
Shi H.P., Gap solitons in periodic discrete Schrödinger equations with nonlinearity. Acta Appl. Math., 2010, 109(3): 1065–1075
Shi H.P., Zhang H.Q., Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J. Math. Anal. Appl., 2010, 361(2): 411–419
Szulkin A., Weth T., Ground state solutions for some indefinite variational problems. J. Funct. Anal., 2009, 257(12): 3802–3822
Tang X.H., Non-Nehari manifold method for superlinear Schrödinger equation. Taiwanese J. Math., 2014, 18(6): 1957–1979
Tang X.H., New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation. Adv. Nonlinear Stud., 2014, 14(2): 361–373
Tang X.H., Non-Nehari manifold method for asymptotically periodic Schrödinger equations. Sci. China Math., 2015, 58(4): 715–728
Tang X.H., Chen S.T., Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differential Equations, 2017, 56 (4): Paper No. 110, 25 pp.
Tang X.H., Chen S.T., Lin X.Y., Yu J.S., Ground state solutions of Nehari–Pankov type for Schrödinger equations with local super-quadratic conditions. J. Differential Equations, 2020, 268(8): 4663–4690
Yang M.B., Chen W.X., Ding Y.H., Solutions for discrete periodic Schrödinger equations with spectrum 0. Acta Appl. Math., 2010, 110(3): 1475–1488
Zhang G., Pankov A., Standing waves of the discrete nonlinear Schrödinger equations with growing potentials. Commun. Math. Anal., 2008, 5(2): 38–49
Zhang G., Pankov A., Standing wave solutions of the discrete nonlinear Schrödinger equations with unbounded potentials. Appl. Anal., 2010, 89(9): 1541–1557
Zhang J., Chen J.H., Li Q.Q., Zhang W., Concentration behavior of semiclassical solutions for Hamiltonian elliptic system. Adv. Nonlinear Anal., 2021, 10(1): 233–260
Zhang W., Chen J., Mi H.L., Ground states and multiple solutions for Hamiltonian elliptic system with gradient term. Adv. Nonlinear Anal., 2021, 10(1): 331–352
Zhou Z., Yu J.S., On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. J. Differential Equations, 2010, 249(5): 1199–1212
Zhou Z., Yu J.S., Chen Y.M., On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity. Nonlinearity, 2010, 23(7): 1727–1740
Zhou Z., Yu J.S., Chen Y.M., Homoclinic solutions in periodic difference equations with saturable nonlinearity. Sci. China Math., 2011, 54(1): 83–93
Acknowledgements
The authors are grateful to the anonymous referees for their contribution to improve the quality of the manuscript. This work was supported in part by the National Natural Science Foundation of China (No. 11301297) and Natural Science Foundation of Hubei Province and Yichang City.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Rights and permissions
About this article
Cite this article
Chen, P., Meng, L. & Tang, X. Ground States for DNLS Equation with Periodic or Asymptotically Periodic Potential. Front. Math 19, 467–494 (2024). https://doi.org/10.1007/s11464-021-0271-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-021-0271-8