## Homoclinic solutions of periodic discrete Schrödinger equations with local superquadratic conditions.(English)Zbl 07504991

### MSC:

 35Q51 Soliton equations 35Q55 NLS equations (nonlinear Schrödinger equations) 39A12 Discrete version of topics in analysis 39A70 Difference operators
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 [1] N. Ackermann and J. Chagoya, Ground states for irregular and indefinite superlinear Schrödinger equations, J. Differential Equations, 261 (2016), pp. 5180-5201. · Zbl 1347.35114 [2] S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, Phys. D, 103 (1997), pp. 201-250. · Zbl 1194.34059 [3] Z. Balanov, C. Garcia-Azpeitia, and W. Krawcewicz, On variational and topological methods in nonlinear difference equations, Commun. Pure Appl. Anal., 17 (2018), pp. 2813-2844. · Zbl 1398.39004 [4] T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279 (2006), pp. 1267-1288. · Zbl 1117.58007 [5] G. Chen and S. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Appl. Math. Comput., 218 (2012), pp. 5496-5507. · Zbl 1254.39006 [6] G. Chen and S. Ma, Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms, Appl. Math. Comput., 232 (2014), pp. 787-798. · Zbl 1410.39008 [7] G. Chen, S. Ma, and Z.-Q. Wang, Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities, J. Differential Equations, 261 (2016), pp. 3493-3518. · Zbl 1360.35218 [8] G. Chen and M. Schechter, Multiple solutions for non-periodic Schrödinger lattice systems with perturbation and super-linear terms, Z. Angew. Math. Phys., 70 (2019), 152. · Zbl 1431.35167 [9] W. Chen and M. Yang, Standing waves for periodic discrete nonlinear Schrödinger equations with asymptotically linear terms, Acta Math. Appl. Sinica. Engl. Ser., 28 (2012), pp. 351-360. · Zbl 1359.35175 [10] D. N. Christodoulides, F. Lederer, and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature, 424 (2003), pp. 817-823. [11] V. Coti Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $$\mathbb{R}^N$$, Commun. Pure Appl. Math., 101 (1992), pp. 1217-1269. · Zbl 0785.35029 [12] Y. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222 (2006), pp. 137-163. · Zbl 1090.35077 [13] L. Erbe, B. Jia, and Q. Zhang, Homoclinic solutions of discrete nonlinear systems via variational method, J. Appl. Anal. Comput., 9 (2019), pp. 271-294. · Zbl 1462.39003 [14] S. Flach and A. V. Gorbach, Discrete breathers-Advance in theory and applications, Phys. Rep., 467 (2008), pp. 1-116. [15] J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of discrete solitons in optically induced real time waveguide arrays, Phys. Rev. Lett., 90 (2003), 023902. [16] J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature, 422 (2003), pp. 147-150. [17] A. V. Gorbach and M. Johansson, Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model, Eur. Phys. J. D, 29 (2004), pp. 77-93. [18] G. Kopidakis, S. Aubry, and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), 165501. [19] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, 1964. [20] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), pp. 441-472. · Zbl 0947.35061 [21] J. Kuang and Z. Guo, Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities, Nonlinear Anal., 89 (2013), pp. 208-218. · Zbl 1325.39004 [22] G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), pp. 763-776. · Zbl 1056.35065 [23] G. Li and C. Wang, The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition, Ann. Acad. Sci. Fenn. Math., 36 (2011), pp. 461-480. · Zbl 1234.35095 [24] G. Lin, J. Yu, and Z. Zhou, Homoclinic solutions of discrete nonlinear Schrödinger equations with partially sublinear nonlinearities, Electron. J. Differential Equations, 96 (2019), pp. 1-14. · Zbl 1417.39027 [25] G. Lin and Z. Zhou, Homoclinic solutions in periodic difference equations with mixed nonlinearities, Math. Methods Appl. Sci., 39 (2016), pp. 245-260. · Zbl 1348.39003 [26] G. Lin and Z. Zhou, Homoclinic solutions in non-periodic discrete $$\phi$$-Laplacian equations with mixed nonlinearities, Appl. Math. Lett., 64 (2017), pp. 15-20. · Zbl 1353.39007 [27] G. Lin and Z. Zhou, Homoclinic solutions of discrete $$\phi$$-Laplacian equations with mixed nonlinearities, Commun. Pure Appl. Anal., 17 (2018), pp. 1723-1747. · Zbl 1396.39008 [28] G. Lin, Z. Zhou, and J. Yu, Ground state solutions of discrete asymptotically linear Schrödinger equations with bounded and non-periodic potentials, J. Dynam. Differential Equations, 32 (2020), pp. 527-555. · Zbl 1439.39010 [29] S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), pp. 1-9. · Zbl 1247.35149 [30] Z. Liu, J. Su, and T. Weth, Compactness results for Schrödinger equations with asymptotically linear terms, J. Differential Equations, 231 (2006), pp. 501-512. · Zbl 1387.35246 [31] R. Livi, R. Franzosi, and G.-L. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), 060401. [32] S. Ma and Z. Wang, Multibump solutions for discrete periodic nonlinear Schrödinger equations, Z. Angew. Math. Phys., 64 (2013), pp. 1413-1442. · Zbl 1280.35139 [33] A. Mai and Z. Zhou, Discrete solitons for periodic discrete nonlinear Schrödinger equations, Appl. Math. Comput., 222 (2013), pp. 34-41. · Zbl 1329.35285 [34] A. Mai and Z. Zhou, Ground state solutions for the periodic discrete nonlinear Schrödinger equations with superlinear nonlinearities, Abstr. Appl. Anal., 2013 (2013), 317139. · Zbl 1291.35356 [35] J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Commun. Partial Differential Equations, 41 (2016), pp. 1426-1440. · Zbl 1353.35267 [36] A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), pp. 259-287. · Zbl 1225.35222 [37] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19 (2006), pp. 27-40. · Zbl 1220.35163 [38] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations II. A generalized Nehari manifold approach, Discrete Contin. Dyn. Syst., 19 (2007), pp. 419-430. · Zbl 1220.35164 [39] A. Pankov and V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), pp. 3219-3236. · Zbl 1186.35206 [40] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, J. Math. Anal. Appl., 371 (2010), pp. 254-265. · Zbl 1197.35273 [41] F. Pavia, W. Kryszewski, and A. Szulkin, Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term, Proc. Amer. Math. Soc., 145 (2017), pp. 4783-4794. · Zbl 1375.35119 [42] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, AMS, Providence, RI, 1986. · Zbl 0609.58002 [43] M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var., 9 (2003), pp. 601-619. · Zbl 1173.35482 [44] H. Shi and H. Zhang, Existence of gap solitons in periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361 (2010), pp. 411-419. · Zbl 1178.35351 [45] M. Stepic, D. Kip, L. Hadzievski, and A. Maluckov, One-dimensional bright discrete solitons in media with saturable nonlinearity, Phys. Rev. E, 69 (2004), 066618. [46] M. Struwe, Variational Methods, 2nd ed., Springer-Verlag, Berlin, 1996. · Zbl 0864.49001 [47] A. A. Sukhorukov and Y. S. Kivshar, Generation and stability of discrete gap solitons, Opt. Lett., 28 (2003), pp. 2345-2347. [48] J. Sun and S. Ma, Multiple solutions for discrete periodic nonlinear Schrödinger equations, J. Math. Phys., 56 (2015), 022110. · Zbl 1360.81152 [49] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), pp. 3802-3822. · Zbl 1178.35352 [50] X. Tang, Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. Engl. Ser., 32 (2016), pp. 463-473. · Zbl 1386.39015 [51] X. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), pp. 104-116. · Zbl 1314.35030 [52] X. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), pp. 715-728. · Zbl 1321.35055 [53] X. Tang, S. Chen, X. Lin, and J. Yu, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differential Equations, 268 (2020), pp. 4663-4690. · Zbl 1437.35224 [54] X. Tang, X. Lin, and J. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dynam. Differential Equations, 31 (2019), pp. 369-383. · Zbl 1414.35062 [55] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathe. Surveys Monogr. 72, AMS, Providence, RI, 2000. · Zbl 1056.39029 [56] V. O. Vinetskii and N. V. Kukhtarev, Theory of the conductivity induced by recording holographic gratings in nonmetallic crystals, Sov. Phys. Solid State, 16 (1975), pp. 2414-2415. [57] M. Yang, W. Chen, and Y. Ding, Solutions for discrete periodic Schrödinger equations with spectrum 0, Acta Appl. Math., 110 (2010), pp. 1475-1488. · Zbl 1191.35260 [58] J. Yeh, Lectures on Real Analysis, World Scientific, Singapore, 2000. · Zbl 1039.28001 [59] G. Zhang and A. Pankov, Standing waves of discrete nonlinear Schrödinger equations with growing potential, Commun. Math. Anal., 5 (2008), pp. 38-49. · Zbl 1168.35437 [60] G. Zhang, Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials, J. Math. Phys., 50 (2009), 013505. · Zbl 1200.37072 [61] Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions, Commun. Pure Appl. Anal., 18 (2019), pp. 425-434. · Zbl 1401.39017 [62] Z. Zhou and D. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math., 58 (2015), pp. 781-790. · Zbl 1328.39011 [63] Z. Zhou and J. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), pp. 1199-1212. · Zbl 1200.39001 [64] Z. Zhou and J. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Sin. Engl. Ser., 29 (2013), pp 1809-1822. · Zbl 1284.39006 [65] Z. Zhou, J. Yu, and Y. Chen, On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity, Nonlinearity, 23 (2010), pp. 1727-1740. · Zbl 1193.35176 [66] Z. Zhou, J. Yu, and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math., 54 (2011), pp. 83-93. · Zbl 1239.39010 [67] Q. Zhu, Z. Zhou, and L. Wang, Existence and stability of discrete solitons in nonlinear Schrödinger lattices with hard potentials, Phys. D, 403 (2020), 132326. · Zbl 07490558
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