Abstract
In the present paper, we are interested in the existence of least energy sign-changing solutions for a class of discrete Schrödinger equations involving asymptotically linear behavior. When the linear potentials are assumed to be unbounded at the infinities and the nonlinearities satisfy certain strict conditions, a least energy sign-changing solution has been obtained via constraint variational method, degree theory and deformation lemma. Moreover, the solution changes sign exactly once and has a fast exponential decay with any order, which is firstly obtained by a comparison principle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
All data included in this study are available upon request by contacting with the corresponding author.
References
Chen, S., Li, Y., Tang, X.: Sign-changing solutions for asymptotically linear Schrödinger equation in bounded domains. Electron. J. Differ. Equ. 317, 1–9 (2016)
Chen, S., Tang, X., Yu, J.: Sign-changing ground state solutions for discrete nonlinear Schrödinger equations. J. Differ. Equ. Appl. 25(2), 202–218 (2019)
Davydov, A.S.: The theory of contraction of proteins under their excitation. J. Theor. Biol. 38(3), 559–569 (1973)
Davydov, A.S.: Solitons and energy transfer along protein molecules. J. Theor. Biol. 66(2), 379–387 (1977)
Deng, Y., Shuai, W.: Sign-changing solutions for non-local elliptic equations involving the fractional Laplacain. Adv. Differ. Equ. 23(1/2), 109–134 (2018)
Flach, S., Gorbach, A.V.: Discrete breathers-advances in theory and applications. Phys. Rep. 467(1–3), 1–116 (2008)
Hennig, D., Tsironis, G.P.: Wave transmission in nonlinear lattices. Phys. Rep. 307(5–6), 333–432 (1999)
Kopidakis, G., Aubry, S., Tsironis, G.P.: Targeted energy transfer through discrete breathers in nonlinear systems. Phys. Rev. Lett. 87(16), 165501 (2001)
Lin, X., Tang, X.: Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems. J. Math. Anal. Appl. 373(1), 59–72 (2011)
Mai, A., Zhou, Z.: Discrete solitons for periodic discrete nonlinear Schrödinger equations. Appl. Math. Comput. 222, 34–41 (2013)
Miranda C.: Un’osservazione su un teorema di Brouwer. Consiglio Nazionale delle Ricerche (1940)
Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations. Nonlinearity 19(1), 27 (2005)
Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations II: a generalized Nehari manifold approach. Discret. Contin. Dyn. Syst. 19(2), 419 (2007)
Pankov, A.: Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities. J. Math. Anal. Appl. 371(1), 254–265 (2010)
Pankov, A., Zakharchenko, N.: On some discrete variational problems. Acta Applicandae Mathematica 65(1), 295–303 (2001)
Pankov, A., Zhang, G.: Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearity. J. Math. Sci. 177(1), 71–82 (2011)
Shuai, W.: Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. J. Differ. Equ. 259(4), 1256–1274 (2015)
Tang, X.: Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation. Acta Mathematica Sinica, English Series 32(4), 463–473 (2016)
Tang, X., Cheng, B.: Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J. Differ. Equ. 261(4), 2384–2402 (2016)
Teschl G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. American Mathematical Soc. (2000)
Willem, M.: Minimax Theorems. Springer, Berlin (1997)
Yang, M., Chen, W., Ding, Y.: Solutions for discrete periodic Schrödinger equations with spectrum 0. Acta Appl. Math. 110(3), 1475 (2010)
Zhang, G., Pankov, A.: Standing waves of the discrete nonlinear Schrödinger equations with growing potentials. Commun. Math. Anal. 5(2) (2008)
Zhou, Z., Yu, J.: On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. J. Differ. Equ. 249(5), 1199–1212 (2010)
Zhou, Z., Yu, J.S.: Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity. Acta Mathematica Sinica, English Series 29(9), 1809–1822 (2013)
Zhou, Z., Yu, J., Chen, Y.: On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity. Nonlinearity 23(7), 1727 (2010)
Funding
This study was funded by the Natural Science Foundation of Guangdong Province (Nos. 2021A1515010383, 2022A1515010644) and the Project of Science and Technology of Guangzhou (No. 202102020730).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We declare that we have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research is supported by the Natural Science Foundation of Guangdong Province (Nos. 2021A1515010383, 2022A1515010644), the Project of Science and Technology of Guangzhou (No. 202102020730).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Fan, Y., Xie, Q. Sign-Changing Solutions for Discrete Schrödinger Equations with Asymptotically Linear Term. Results Math 78, 243 (2023). https://doi.org/10.1007/s00025-023-02018-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-02018-x