Solutions to discrete nonlinear Kirchhoff-Choquard equations. (English) Zbl 07909827
Summary: In this paper, we study the discrete Kirchhoff-Choquard equation
\[
-\left( a+b \int_{\mathbb{Z}^3}|\nabla u|^2 d\mu \right) \Delta u+V(x) u=\left( R_{\alpha} *F(u)\right) f(u),\quad x\in\mathbb{Z}^3,
\]
where \(a,\,b>0, \alpha \in (0,3)\) are constants and \(R_{\alpha}\) is the Green’s function of the discrete fractional Laplacian that behaves as the Riesz potential. Under some suitable assumptions on \(V\) and \(f\), we prove the existence of nontrivial solutions and ground state solutions respectively by variational methods.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
References:
| [1] | Benchira, H., Matallah, A., El Mokhtar, M.E.O., Sabri, K.: The existence result for a \(p\)-Kirchhoff-type problem involving critical Sobolev exponent. J. Funct. Spaces, Art. ID 3247421, 8 pp (2023) · Zbl 07904085 |
| [2] | Chen, S., Tang, X.: Berestycki-Lions conditions on ground state solutions for Kirchhoff-type problems with variable potentials. J. Math. Phys. 60(12), 121509, 16 pp (2019) · Zbl 1432.35192 |
| [3] | Chen, S.; Zhang, B.; Tang, X., Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal., 9, 148-167, 2020 · Zbl 1421.35100 · doi:10.1515/anona-2018-0147 |
| [4] | Cheng, B.; Wu, X., Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal., 71, 4883-4892, 2009 · Zbl 1175.35038 · doi:10.1016/j.na.2009.03.065 |
| [5] | Eddine, NC; Nguyen, AT; Ragusa, MA, The Dirichlet problem for a class of anisotropic Schrödinger-Kirchhoff-type equations with critical exponent, Math. Model. Anal., 29, 254-267, 2024 · Zbl 1542.35037 · doi:10.3846/mma.2024.19006 |
| [6] | El-Houari, H.; Chadli, LS; Moussa, H., Multiple solutions in fractional Orlicz-Sobolev spaces for a class of nonlocal Kirchhoff systems, Filomat, 38, 2857-2875, 2024 |
| [7] | Grigor’yan, A.; Lin, Y.; Yang, Y., Existence of positive solutions to some nonlinear equations on locally finite graphs, Sci. China Math., 60, 1311-1324, 2017 · Zbl 1384.34035 · doi:10.1007/s11425-016-0422-y |
| [8] | Gu, G.; Tang, X., The concentration behavior of ground states for a class of Kirchhoff-type problems with Hartree-type nonlinearity, Adv. Nonlinear Stud., 19, 779-795, 2019 · Zbl 1427.35050 · doi:10.1515/ans-2019-2045 |
| [9] | Guariglia, E.: Riemann zeta fractional derivative-functional equation and link with primes. Adv. Differ. Equ., Paper No. 261, 15 pp (2019) · Zbl 1459.26011 |
| [10] | Guariglia, E., Fractional calculus, zeta functions and Shannon entropy, Open Math., 19, 1, 87-100, 2021 · Zbl 1475.11151 · doi:10.1515/math-2021-0010 |
| [11] | Guo, Z., Ground states for Kirchhoff equations without compact condition, J. Differ. Equ., 259, 2884-2902, 2015 · Zbl 1319.35018 · doi:10.1016/j.jde.2015.04.005 |
| [12] | Han, X.; Shao, M.; Zhao, L., Existence and convergence of solutions for nonlinear biharmonic equations on graphs, J. Differ. Equ., 268, 3936-3961, 2020 · Zbl 1433.35049 · doi:10.1016/j.jde.2019.10.007 |
| [13] | He, F.; Qin, D.; Tang, X., Existence of ground states for Kirchhoff-type problems with general potentials, J. Geom. Anal., 31, 7709-7725, 2021 · Zbl 1473.35265 · doi:10.1007/s12220-020-00546-4 |
| [14] | He, X.; Zou, W., Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({\mathbb{R} }^3\), J. Differ. Equ., 252, 1813-1834, 2012 · Zbl 1235.35093 · doi:10.1016/j.jde.2011.08.035 |
| [15] | Hu, D., Tang, X., Yuan, S., Zhang, Q.: Ground state solutions for Kirchhoff-type problems with convolution nonlinearity and Berestycki-Lions type conditions. Anal. Math. Phys. 12(1), Paper No. 19, 27 pp (2022) · Zbl 1481.35211 |
| [16] | Hu, D., Tang, X., Zhang, N.: Semiclassical ground state solutions for a class of Kirchhoff-Type problem with convolution nonlinearity. J. Geom. Anal. 32(11), Paper No. 272, 37 pp (2022) · Zbl 1497.35229 |
| [17] | Hua, B.; Li, R.; Wang, L., A class of semilinear elliptic equations on groups of polynomial growth, J. Differ. Equ., 363, 327-349, 2023 · Zbl 1515.35106 · doi:10.1016/j.jde.2023.03.025 |
| [18] | Hua, B., Xu, W.: Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs. Calc. Var. Partial Differ. Equ. 62(4), Paper No. 127, 17 pp (2023) · Zbl 1511.35325 |
| [19] | Huang, G.; Li, C.; Yin, X., Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 35, 3, 935-942, 2015 · Zbl 1304.26020 · doi:10.3934/dcds.2015.35.935 |
| [20] | Li, C.; Dao, X.; Guo, P., Fractional derivatives in complex planes, Nonlinear Anal., 71, 1857-1869, 2009 · Zbl 1173.26305 · doi:10.1016/j.na.2009.01.021 |
| [21] | Li, G.; Ye, H., Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \({\mathbb{R} }^3\), J. Differ. Equ., 257, 566-600, 2014 · Zbl 1290.35051 · doi:10.1016/j.jde.2014.04.011 |
| [22] | Li, R., Wang, L.: The existence and convergence of solutions for the nonlinear Choquard equations on groups of polynomial growth. arXiv: 2208.00236 |
| [23] | Liang, S.; Sun, M.; Shi, S.; Liang, S., On multi-bump solutions for the Choquard-Kirchhoff equations in \({\mathbb{R} }^N\), Discrete Contin. Dyn. Syst. Ser. S, 16, 3163-3193, 2023 · Zbl 1534.35218 · doi:10.3934/dcdss.2023012 |
| [24] | Liang, S.; Pucci, P.; Zhang, B., Multiple solutions for critical Choquard-Kirchhoff type equations, Adv. Nonlinear Anal., 10, 400-419, 2021 · Zbl 1465.35215 · doi:10.1515/anona-2020-0119 |
| [25] | Liu, Y., Zhang, M.: Existence of solutions for nonlinear biharmonic Choquard equations on weighted lattice graphs. J. Math. Anal. Appl. 534(2), Paper No. 128079, 18 pp (2024) · Zbl 1533.35100 |
| [26] | Liu, Y., Zhang, M.: The ground state solutions to a class of biharmonic Choquard equations on weighted lattice graphs. Bull. Iran. Math. Soc. 50(1), Paper No. 12, 17 pp (2024) · Zbl 1541.35527 |
| [27] | Lü, D., A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99, 35-48, 2014 · Zbl 1286.35108 · doi:10.1016/j.na.2013.12.022 |
| [28] | Lü, D., Dai, S.: Existence and asymptotic behavior of solutions for Kirchhoff equations with general Choquard-type nonlinearities. Z. Angew. Math. Phys. 74(6), Paper No. 232, 15 pp (2023) · Zbl 1529.35238 |
| [29] | Lü, D.; Lu, Z., On the existence of least energy solutions to a Kirchhoff-type equation in \({\mathbb{R} }^3\), Appl. Math. Lett., 96, 179-186, 2019 · Zbl 1427.35058 · doi:10.1016/j.aml.2019.04.028 |
| [30] | Lü, W., Ground states of a Kirchhoff equation with the potential on the lattice graphs, Commun. Anal. Mech., 15, 792-810, 2023 · Zbl 07919376 · doi:10.3934/cam.2023038 |
| [31] | Michelitsch, T., Collet, B., Riascos, A. Nowakowski, A., Nicolleau, F.: Recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices. J. Phys. A 50(50), 505004, 29 pp (2017) · Zbl 1394.82009 |
| [32] | Ragusa, MA, Commutators of fractional integral operators on vanishing-Morrey spaces, J. Global Optim., 40, 361-368, 2008 · Zbl 1143.42020 · doi:10.1007/s10898-007-9176-7 |
| [33] | Ragusa, MA, Parabolic Herz spaces and their applications, Appl. Math. Lett., 25, 10, 1270-1273, 2012 · Zbl 1255.35131 · doi:10.1016/j.aml.2011.11.022 |
| [34] | Sun, D.; Zhang, Z., Uniqueness, existence and concentration of positive ground state solutions for Kirchhoff type problems in \({\mathbb{R} }^3\), J. Math. Anal. Appl., 461, 128-149, 2018 · Zbl 1435.35008 · doi:10.1016/j.jmaa.2018.01.003 |
| [35] | Szulkin, A.; Weth, T., The method of Nehari manifold. Handbook of nonconvex analysis and applications, 597-632, 2010, Somerville: International Press, Somerville · Zbl 1218.58010 |
| [36] | Tang, X., Chen, S.: Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differ. Equ. 56(4), Paper No. 110, 25 pp (2017) · Zbl 1376.35056 |
| [37] | Wang, L.: The ground state solutions to discrete nonlinear Choquard equations with Hardy weights. Bull. Iran. Math. Soc. 49(3), Paper No. 30, 29 pp (2023) · Zbl 1518.35346 |
| [38] | Wang, L.: The ground state solutions of discrete nonlinear Schrödinger equations with Hardy weights. Mediterr. J. Math. 21(3), Paper No. 78 (2024) |
| [39] | Willem, M.: Minimax theorems. In: Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser, Boston (1996) · Zbl 0856.49001 |
| [40] | Wu, X., Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in \({\mathbb{R} }^N\), Nonlinear Anal. Real World Appl., 12, 1278-1287, 2011 · Zbl 1208.35034 · doi:10.1016/j.nonrwa.2010.09.023 |
| [41] | Wu, M.; Tang, C., The existence and concentration of ground state sign-changing solutions for Kirchhoff-type equations with a steep potential well, Acta Math. Sci. Ser. B (Engl. Ed.), 43, 1781-1799, 2023 · Zbl 1524.35202 |
| [42] | Yin, L., Gan, W., Jiang, S.: Existence and concentration of ground state solutions for critical Kirchhoff-type equation invoLüing Hartree-type nonlinearities. Z. Angew. Math. Phys. 73(3), Paper No. 103, 19 pp (2022) · Zbl 1490.35172 |
| [43] | Zhang, N.; Zhao, L., Convergence of ground state solutions for nonlinear Schrödinger equations on graphs, Sci. China Math., 61, 1481-1494, 2018 · Zbl 1401.35093 · doi:10.1007/s11425-017-9254-7 |
| [44] | Zhou, L., Zhu, C.: Ground state solution for a class of Kirchhoff-type equation with general convolution nonlinearity. Z. Angew. Math. Phys. 73(2), Paper No. 75, 13 pp (2022) · Zbl 1486.35220 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
