Multiplicity results for discrete partial mean curvature problems. (English) Zbl 07830969
Summary: In this article we continue the study of nonlinear discrete Dirichlet boundary value problems driven by mean curvature operators. By using a consequence of the local minimum theorem due Bonanno and mountain pass theorem, we will obtain a new multiplicity results of the solutions for a nonlinear discrete Dirichlet boundary value problems driven by \(\phi_c\)-Laplacian operator which have applications in the dynamic model of combustible gases, the capillarity problem in hydrodynamics, and flux-limited diffusion phenomenon, under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Also, we establish the existence of third solution for our problem, by mountain pass theorem given by Pucci and Serrin.
MSC:
| 39A23 | Periodic solutions of difference equations |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
Keywords:
multiple solutions; \(\phi_c\)-Laplacian boundary value; critical point theory; variational methods; critical point theoryReferences:
| [1] | Ambrosetti, A.; Rabinowitz, PH, Dual variational methods in critical point theory and applications, J Funct Anal, 14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7 |
| [2] | Bereanu, C.; Jebelean, P.; Mawhin, J., Multiple solutions for Neumann and periodic problems with singular \(\phi \)-Laplacian, J Funct Anal, 261, 3226-3246 (2011) · Zbl 1241.35076 · doi:10.1016/j.jfa.2011.07.027 |
| [3] | Bereanu, C.; Jebelean, P.; Mawhin, J., Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces, Math Nachr, 283, 379-391 (2010) · Zbl 1185.35113 · doi:10.1002/mana.200910083 |
| [4] | Bereanu, C.; Jebelean, P.; Mawhin, J., Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, Calc Var Partial Differ Equ, 46, 113-122 (2013) · Zbl 1262.35088 · doi:10.1007/s00526-011-0476-x |
| [5] | Bereanu, C.; Jebelean, P.; Torres, PJ, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J Funct Anal, 264, 270-287 (2013) · Zbl 1336.35174 · doi:10.1016/j.jfa.2012.10.010 |
| [6] | Bonanno, G., A critical point theorem via the Ekeland variational principle, Nonlinear Anal TMA, 75, 2992-3007 (2012) · Zbl 1239.58011 · doi:10.1016/j.na.2011.12.003 |
| [7] | Bonanno, G.; Candito, P., Infinitely many solutions for a class of discrete nonlinear boundary value problems, Appl Anal, 88, 605-616 (2009) · Zbl 1176.39004 · doi:10.1080/00036810902942242 |
| [8] | Bonanno, G.; Candito, P., Nonlinear difference equations investigated via critical point methods, Nonlinear Anal, 70, 3180-3186 (2009) · Zbl 1166.39006 · doi:10.1016/j.na.2008.04.021 |
| [9] | Bonanno, G.; Candito, P.; D’Aguì, G., Positive solutions for a nonlinear parameter-depending algebraic system, Electron J Differ Equ, 2015, 1-14 (2015) · Zbl 1310.47112 |
| [10] | Bonanno, G.; Jebelean, P.; Şerban, C., Superlinear discrete problems, Appl Math Lett, 52, 162-168 (2016) · Zbl 1330.39001 · doi:10.1016/j.aml.2015.09.005 |
| [11] | Bonanno, G.; Livrea, R.; Mawhin, J., Existence results for parametric boundary value problems involving the mean curvature operator, Nonlinear Differ Equ Appl, 22, 411-426 (2015) · Zbl 1410.34073 · doi:10.1007/s00030-014-0289-7 |
| [12] | Chen, Y.; Zhou, Z., Existence of three solutions for a nonlinear discrete boundary value problem with \(\varphi_c\)-Laplacian, Symmetry, 12, 1839 (2020) · doi:10.3390/sym12111839 |
| [13] | D’Aguì, G., Multiplicity results for nonlinear mixed boundary value problem, Bound Value Prob, 2012, 134 (2012) · Zbl 1281.34022 · doi:10.1186/1687-2770-2012-134 |
| [14] | Du, S.; Zhou, Z., Multiple solutions for partial discrete Dirichlet problems involving the \(p\)-Laplacian, Mathematics, 8, 20-30 (2020) · doi:10.3390/math8112030 |
| [15] | Du, S.; Zhou, Z., On the existence of multiple solutions for a partial discrete Dirichlet boundary value problem with mean curvature operator, Adv Nonlinear Anal, 11, 198-211 (2022) · Zbl 1473.39012 · doi:10.1515/anona-2020-0195 |
| [16] | Ecker, K.; Huisken, G., Interior estimates for hypersurfaces moving by mean-curvarture, Invent Math, 150, 547-569 (1991) · Zbl 0707.53008 · doi:10.1007/BF01232278 |
| [17] | Ecker, K.; Huisken, G., Mean curvature evolution of entire graphs, Ann Math, 130, 453-471 (1989) · Zbl 0696.53036 · doi:10.2307/1971452 |
| [18] | Finn, R., Equilibrium capillary surfaces (1986), New York: Springer, New York · Zbl 0583.35002 · doi:10.1007/978-1-4613-8584-4 |
| [19] | Galewski, M.; Gła̧b, S.; Wieteska, R., Positive solutions for anisotropic discrete boundary value problems, Electron J Differ Equ, 2013, 1-9 (2013) · Zbl 1321.39004 |
| [20] | Galewski, M.; Heidarkhani, S.; Salari, A., Multiplicity results for discrete anisotropic equations, Discrete Contin Dyn Syst Ser, 23, 203-218 (2018) · Zbl 1374.34061 |
| [21] | Galewski, M.; Orpel, A., On the existence of solutions for discrete elliptic boundary value problems, Appl Anal, 89, 1879-1891 (2010) · Zbl 1204.35096 · doi:10.1080/00036811.2010.499508 |
| [22] | Ghobadi A, Heidarkhani S (2023) Critical point approaches for doubly eigenvalue discrete boundary value problems driven by \(\phi_c\)-Laplacian operator, submitted |
| [23] | Graef, JR; Heidarkhani, S.; Kong, L., Nontrivial periodic solutions to second-order impulsive Hamiltonian systems, Electron J Differ Equ, 2015, 204, 1-17 (2015) · Zbl 1329.34085 |
| [24] | He, TS; Zhou, YW; Xu, YT; Chen, CY, Sign-changing solutions for discrete second-order periodic boundary value problems, Bull Malays Math Sci Soc, 38, 181-195 (2015) · Zbl 1308.39003 · doi:10.1007/s40840-014-0012-1 |
| [25] | Heidarkhani, S.; Afrouzi, GA; Caristi, G.; Henderson, J.; Moradi, S., A variational approach to difference equations, J Differ Equ Appl, 22, 1761-1776 (2016) · Zbl 1375.39013 · doi:10.1080/10236198.2016.1243671 |
| [26] | Heidarkhani, S.; Imbesi, M., Multiple solutions for partial discrete Dirichlet problems depending on a real parameter, J Differ Equ Appl, 21, 96-110 (2015) · Zbl 1320.39007 · doi:10.1080/10236198.2014.988619 |
| [27] | Kurganov, A.; Rosenau, P., On reaction processes with saturating diffusion, Nonlinearity, 19, 171-193 (2006) · Zbl 1094.35063 · doi:10.1088/0951-7715/19/1/009 |
| [28] | Obersnel, F.; Omari, P., Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, J Differ Equ, 249, 1674-1725 (2010) · Zbl 1207.35145 · doi:10.1016/j.jde.2010.07.001 |
| [29] | Obersnel, F.; Omari, P.; Rivetti, S., Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions, Nonlinear Anal RWA, 13, 2830-2852 (2012) · Zbl 1279.34054 · doi:10.1016/j.nonrwa.2012.04.012 |
| [30] | Pucci, P.; Serrin, J., A mountain pass theorem, J Differ Equ, 60, 142-149 (1985) · Zbl 0585.58006 · doi:10.1016/0022-0396(85)90125-1 |
| [31] | Rabinowitz, PH, Minimax methods in critical point theory with applications to differential equations, CBMS regional conference series in mathematics (1986), Providence: The American Mathematical Society, Providence · Zbl 0609.58002 · doi:10.1090/cbms/065 |
| [32] | Rey, O., Heat flow for the equation of surfaces with prescribed mean curvature, Math Ann, 291, 123-146 (1991) · Zbl 0761.58052 · doi:10.1007/BF01445195 |
| [33] | Zhong, T.; Guochun, W., On the heat flow equation of surfaces of constant mean curvature in higher dimensions, Acta Math Sci, 31, 1741-1748 (2011) · Zbl 1249.35130 · doi:10.1016/S0252-9602(11)60358-5 |
| [34] | Zhou, Z.; Ling, J., Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with \(\phi_c\)-Laplacian, Appl Math Lett, 91, 28-34 (2019) · Zbl 1409.39004 · doi:10.1016/j.aml.2018.11.016 |
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.
