# American Institute of Mathematical Sciences

January  2018, 23(1): 203-218. doi: 10.3934/dcdsb.2018014

## Multiplicity results for discrete anisotropic equations

 1 Institute of Mathematics, Technical University of Lodz, Wolczanska 215, 90-924 Lodz, Poland 2 Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran 3 Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran

* Corresponding authorr: Marek Galewski

Received  November 2016 Published  January 2018

In this article we continue the study of discrete anisotropic equations and we will provide a new multiplicity results of the solutions for a discrete anisotropic equation. The procedure viewed here is according to variational methods and critical point theory. In fact, using a consequence of the local minimum theorem due Bonanno and mountain pass theorem we look into the existence results for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by mingling two algebraic conditions on the nonlinear term employing two consequences of the local minimum theorem due Bonanno we guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

Citation: Marek Galewski, Shapour Heidarkhani, Amjad Salari. Multiplicity results for discrete anisotropic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 203-218. doi: 10.3934/dcdsb.2018014
##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [2] F. M. Atici and A. Cabada, Existence and uniqueness results for discrete second-order periodic boundary value problems, Comput. Math. Appl., 45 (2003), 1417-1427. doi: 10.1016/S0898-1221(03)00097-X. Google Scholar [3] F. M. Atici and G. Sh. Guseinov, Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. Math. Anal. Appl., 232 (1999), 166-182. doi: 10.1006/jmaa.1998.6257. Google Scholar [4] M. Bendahmane and K. H. Karlsen, Renormalized solutions of an anisotropic reactiondiffusion-advection system with L1-data, Commun. Pure Appl. Anal., 5 (2006), 733-762. doi: 10.3934/cpaa.2006.5.733. Google Scholar [5] M. Bendahmane, M. Langlais and M. Saad, On some anisotropic reaction-diffusion systems with L1-data modeling the propagation of an epidemic disease, Nonlinear Anal. TMA, 54 (2003), 617-636. doi: 10.1016/S0362-546X(03)00090-7. Google Scholar [6] M. Bojowald, H. Hernandez and H. Morales-Tecotl, A perturbative degrees of freedom in loop quantum gravity: anisotropies, Class. Quantum Grav., 23 (2006), 3491-3516. doi: 10.1088/0264-9381/23/10/017. Google Scholar [7] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. TMA, 75 (2012), 2992-3007. doi: 10.1016/j.na.2011.12.003. Google Scholar [8] G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal. TMA, 70 (2009), 3180-3186. doi: 10.1016/j.na.2008.04.021. Google Scholar [9] G. Bonanno, P. Jebelean and C. Serban, Three solutions for discrete anisotropic periodic and Neumann problems, Dynamic Sys. Appl., 22 (2013), 183-196. Google Scholar [10] A. Cabada, A. Iannizzotto and S. Tersian, Multiple solutions for discrete boundary value problem, J. Math. Anal. Appl., 356 (2009), 418-428. doi: 10.1016/j.jmaa.2009.02.038. Google Scholar [11] P. Candito and N. Giovannelli, Multiple solutions for a discrete boundary value problem, Comput. Math. Appl., 56 (2008), 959-964. doi: 10.1016/j.camwa.2008.01.025. Google Scholar [12] J. Chu and D. Jiang, Eigenvalues and discrete boundary value problems for the onedimensional $p$-Laplacian, J. Math. Anal. Appl., 305 (2005), 452-465. doi: 10.1016/j.jmaa.2004.10.055. Google Scholar [13] G. D'Aguì, Multiplicity results for nonlinear mixed boundary value problem, Bound. Value Probl., 2012 (2012), 1-12. doi: 10.1186/1687-2770-2012-134. Google Scholar [14] E. Eisenriegler, Anisotropic colloidal particles in critical fluids, J. Chem. Phys. 121 (2004), p3299. doi: 10.1063/1.1768514. Google Scholar [15] A. El Hamidi and J. Vétois, Sharp Sobolev asymptotics for critical anisotropic equations, Arch. Ration. Mech. Anal., 192 (2009), 1-36. doi: 10.1007/s00205-008-0122-8. Google Scholar [16] I. Fragala, F. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 715-734. doi: 10.1016/j.anihpc.2003.12.001. Google Scholar [17] H. Gajewski, K. Groeger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1974. Google Scholar [18] M. Galewski and S. Głąb, On the discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl., 386 (2012), 956-965. doi: 10.1016/j.jmaa.2011.08.053. Google Scholar [19] M. Galewski, S. Głąb and R. Wieteska, Positive solutions for anisotropic discrete boundary value problems, Electron. J. Differ. Equ., 2013 (2013), 1-9. Google Scholar [20] M. Galewski and R. Wieteska, Existence and multiplicity of positive solutions for discrete anisotropic equations, Turk. J. Math., 38 (2014), 297-310. doi: 10.3906/mat-1303-6. Google Scholar [21] M. Galewski and R. Wieteska, On the system of anisotropic discrete BVPs, J. Differ. Equ. Appl., 19 (2013), 1065-1081. doi: 10.1080/10236198.2012.709508. Google Scholar [22] J. Garnier, High-frequency asymptotics for Maxwell's equations in anisotropic media, Part Ⅰ: linear geometric and diffractive optics, J. Math. Phys., 42 (2001), 1612-1635. doi: 10.1063/1.1354639. Google Scholar [23] J. Garnier, High-frequency asymptotics for Maxwell's equations in anisotropic media, Part Ⅱ: nonlinear propagation and frequency conversion, J. Math. Phys., 42 (2001), 1636-1654. doi: 10.1063/1.1354640. Google Scholar [24] S. Heidarkhani and M. Khaleghi Moghadam, Existence of three solutions for perturbed nonlinear difference equations, Opuscula Math., 34 (2014), 747-761. doi: 10.7494/OpMath.2014.34.4.747. Google Scholar [25] S. Heidarkhani, M. Ferrara, A. Salari and G. Caristi, Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions, Compl. Var. Ellipt. Equ., 61 (2016), 1494-1516. doi: 10.1080/17476933.2016.1182520. Google Scholar [26] S. Heidarkhani and A. Salari, Nontrivial solutions for impulsive fractional differential systems through variational methods, Comput. Math. Appl. (2016). doi: 10.1016/j.camwa.2016.04.016. Google Scholar [27] L. Jiang and Z. Zhou, Three solutions to Dirichlet boundary value problems for $p$-Laplacian difference equations, Adv. Differ. Equ., 2008 (2008), Art. ID 345916, 10 pp. Google Scholar [28] W. G. Kelly and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, New York, Basel, 1991. Google Scholar [29] M. Khaleghi Moghadam, S. Heidarkhani and J. Henderson, Infinitely many solutions for perturbed difference equations, J. Differ. Equ. Appl., 20 (2014), 1055-1068. doi: 10.1080/10236198.2014.884219. Google Scholar [30] A. Kristály, M. Mihailescu and V. Rădulescu, Discrete boundary value problems involving oscillatory nonlinearities: small and large solutions, J. Differ. Equ. Appl., 17 (2011), 1431-1440. doi: 10.1080/10236190903555245. Google Scholar [31] H. Liang and P. Weng, Existence and multiple solutions for a second-order difference boundary value problem via critical point theory, J. Math. Anal. Appl., 326 (2007), 511-520. doi: 10.1016/j.jmaa.2006.03.017. Google Scholar [32] P. Lindqvist, On the equation $div\left( {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) + \lambda {\left| u \right|^{p - 2}}u = 0$, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.1090/S0002-9939-1990-1007505-7. Google Scholar [33] M. Mihailescu, P. Pucci and V. Rădulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C.R. Acad. Sci. Paris, Ser. I, 345 (2007), 561-566. doi: 10.1016/j.crma.2007.10.012. Google Scholar [34] M. Mihailescu, P. Pucci and V. Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340 (2008), 687-698. doi: 10.1016/j.jmaa.2007.09.015. Google Scholar [35] M. Mihailescu, V. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems, J. Differ. Equ. Appl., 15 (2009), 557-567. doi: 10.1080/10236190802214977. Google Scholar [36] G. Molica Bisci and D. Repovš, Existence of solutions for p-Laplacian discrete equations, Appl. Math. Comput., 242 (2014), 454-461. doi: 10.1016/j.amc.2014.05.118. Google Scholar [37] G. Molica Bisci and D. Repovš, On sequences of solutions for discrete anisotropic equations, Expo. Math., 32 (2014), 284-295. doi: 10.1016/j.exmath.2013.12.001. Google Scholar [38] P. Pucci and J. Serrin, A mountain pass theorem, J. Differ. Eqs., 60 (1985), 142-149. doi: 10.1016/0022-0396(85)90125-1. Google Scholar [39] P. Pucci and J. Serrin, Extensions of the mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210. doi: 10.1016/0022-1236(84)90072-7. Google Scholar [40] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. Math. Soc. Providence, RI, 1986. doi: 10.1090/cbms/065. Google Scholar [41] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410. doi: 10.1016/S0377-0427(99)00269-1. Google Scholar [42] R. Stegliński, On sequences of large solutions for discrete anisotropic equations, Electron. J. Qual. Theory Differ. Equ., 25 (2015), 1-10. Google Scholar [43] D. B. Wang and W. Guan, Three positive solutions of boundary value problems for pLaplacian difference equations, Comput. Math. Appl., 55 (2008), 1943-1949. doi: 10.1016/j.camwa.2007.08.033. Google Scholar [44] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, 1998. Google Scholar [45] P. J. Y. Wong and L. Xie, Three symmetric solutions of Lidstone boundary value problems for difference and partial difference equations, Comput. Math. Appl., 45 (2003), 1445-1460. doi: 10.1016/S0898-1221(03)00102-0. Google Scholar [46] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅱ/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0. Google Scholar

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [2] F. M. Atici and A. Cabada, Existence and uniqueness results for discrete second-order periodic boundary value problems, Comput. Math. Appl., 45 (2003), 1417-1427. doi: 10.1016/S0898-1221(03)00097-X. Google Scholar [3] F. M. Atici and G. Sh. Guseinov, Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. Math. Anal. Appl., 232 (1999), 166-182. doi: 10.1006/jmaa.1998.6257. Google Scholar [4] M. Bendahmane and K. H. Karlsen, Renormalized solutions of an anisotropic reactiondiffusion-advection system with L1-data, Commun. Pure Appl. Anal., 5 (2006), 733-762. doi: 10.3934/cpaa.2006.5.733. Google Scholar [5] M. Bendahmane, M. Langlais and M. Saad, On some anisotropic reaction-diffusion systems with L1-data modeling the propagation of an epidemic disease, Nonlinear Anal. TMA, 54 (2003), 617-636. doi: 10.1016/S0362-546X(03)00090-7. Google Scholar [6] M. Bojowald, H. Hernandez and H. Morales-Tecotl, A perturbative degrees of freedom in loop quantum gravity: anisotropies, Class. Quantum Grav., 23 (2006), 3491-3516. doi: 10.1088/0264-9381/23/10/017. Google Scholar [7] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. TMA, 75 (2012), 2992-3007. doi: 10.1016/j.na.2011.12.003. Google Scholar [8] G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal. TMA, 70 (2009), 3180-3186. doi: 10.1016/j.na.2008.04.021. Google Scholar [9] G. Bonanno, P. Jebelean and C. Serban, Three solutions for discrete anisotropic periodic and Neumann problems, Dynamic Sys. Appl., 22 (2013), 183-196. Google Scholar [10] A. Cabada, A. Iannizzotto and S. Tersian, Multiple solutions for discrete boundary value problem, J. Math. Anal. Appl., 356 (2009), 418-428. doi: 10.1016/j.jmaa.2009.02.038. Google Scholar [11] P. Candito and N. Giovannelli, Multiple solutions for a discrete boundary value problem, Comput. Math. Appl., 56 (2008), 959-964. doi: 10.1016/j.camwa.2008.01.025. Google Scholar [12] J. Chu and D. Jiang, Eigenvalues and discrete boundary value problems for the onedimensional $p$-Laplacian, J. Math. Anal. Appl., 305 (2005), 452-465. doi: 10.1016/j.jmaa.2004.10.055. Google Scholar [13] G. D'Aguì, Multiplicity results for nonlinear mixed boundary value problem, Bound. Value Probl., 2012 (2012), 1-12. doi: 10.1186/1687-2770-2012-134. Google Scholar [14] E. Eisenriegler, Anisotropic colloidal particles in critical fluids, J. Chem. Phys. 121 (2004), p3299. doi: 10.1063/1.1768514. Google Scholar [15] A. El Hamidi and J. Vétois, Sharp Sobolev asymptotics for critical anisotropic equations, Arch. Ration. Mech. Anal., 192 (2009), 1-36. doi: 10.1007/s00205-008-0122-8. Google Scholar [16] I. Fragala, F. Gazzola and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 715-734. doi: 10.1016/j.anihpc.2003.12.001. Google Scholar [17] H. Gajewski, K. Groeger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1974. Google Scholar [18] M. Galewski and S. Głąb, On the discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl., 386 (2012), 956-965. doi: 10.1016/j.jmaa.2011.08.053. Google Scholar [19] M. Galewski, S. Głąb and R. Wieteska, Positive solutions for anisotropic discrete boundary value problems, Electron. J. Differ. Equ., 2013 (2013), 1-9. Google Scholar [20] M. Galewski and R. Wieteska, Existence and multiplicity of positive solutions for discrete anisotropic equations, Turk. J. Math., 38 (2014), 297-310. doi: 10.3906/mat-1303-6. Google Scholar [21] M. Galewski and R. Wieteska, On the system of anisotropic discrete BVPs, J. Differ. Equ. Appl., 19 (2013), 1065-1081. doi: 10.1080/10236198.2012.709508. Google Scholar [22] J. Garnier, High-frequency asymptotics for Maxwell's equations in anisotropic media, Part Ⅰ: linear geometric and diffractive optics, J. Math. Phys., 42 (2001), 1612-1635. doi: 10.1063/1.1354639. Google Scholar [23] J. Garnier, High-frequency asymptotics for Maxwell's equations in anisotropic media, Part Ⅱ: nonlinear propagation and frequency conversion, J. Math. Phys., 42 (2001), 1636-1654. doi: 10.1063/1.1354640. Google Scholar [24] S. Heidarkhani and M. Khaleghi Moghadam, Existence of three solutions for perturbed nonlinear difference equations, Opuscula Math., 34 (2014), 747-761. doi: 10.7494/OpMath.2014.34.4.747. Google Scholar [25] S. Heidarkhani, M. Ferrara, A. Salari and G. Caristi, Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions, Compl. Var. Ellipt. Equ., 61 (2016), 1494-1516. doi: 10.1080/17476933.2016.1182520. Google Scholar [26] S. Heidarkhani and A. Salari, Nontrivial solutions for impulsive fractional differential systems through variational methods, Comput. Math. Appl. (2016). doi: 10.1016/j.camwa.2016.04.016. Google Scholar [27] L. Jiang and Z. Zhou, Three solutions to Dirichlet boundary value problems for $p$-Laplacian difference equations, Adv. Differ. Equ., 2008 (2008), Art. ID 345916, 10 pp. Google Scholar [28] W. G. Kelly and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, New York, Basel, 1991. Google Scholar [29] M. Khaleghi Moghadam, S. Heidarkhani and J. Henderson, Infinitely many solutions for perturbed difference equations, J. Differ. Equ. Appl., 20 (2014), 1055-1068. doi: 10.1080/10236198.2014.884219. Google Scholar [30] A. Kristály, M. Mihailescu and V. Rădulescu, Discrete boundary value problems involving oscillatory nonlinearities: small and large solutions, J. Differ. Equ. Appl., 17 (2011), 1431-1440. doi: 10.1080/10236190903555245. Google Scholar [31] H. Liang and P. Weng, Existence and multiple solutions for a second-order difference boundary value problem via critical point theory, J. Math. Anal. Appl., 326 (2007), 511-520. doi: 10.1016/j.jmaa.2006.03.017. Google Scholar [32] P. Lindqvist, On the equation $div\left( {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) + \lambda {\left| u \right|^{p - 2}}u = 0$, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.1090/S0002-9939-1990-1007505-7. Google Scholar [33] M. Mihailescu, P. Pucci and V. Rădulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C.R. Acad. Sci. Paris, Ser. I, 345 (2007), 561-566. doi: 10.1016/j.crma.2007.10.012. Google Scholar [34] M. Mihailescu, P. Pucci and V. Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340 (2008), 687-698. doi: 10.1016/j.jmaa.2007.09.015. Google Scholar [35] M. Mihailescu, V. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems, J. Differ. Equ. Appl., 15 (2009), 557-567. doi: 10.1080/10236190802214977. Google Scholar [36] G. Molica Bisci and D. Repovš, Existence of solutions for p-Laplacian discrete equations, Appl. Math. Comput., 242 (2014), 454-461. doi: 10.1016/j.amc.2014.05.118. Google Scholar [37] G. Molica Bisci and D. Repovš, On sequences of solutions for discrete anisotropic equations, Expo. Math., 32 (2014), 284-295. doi: 10.1016/j.exmath.2013.12.001. Google Scholar [38] P. Pucci and J. Serrin, A mountain pass theorem, J. Differ. Eqs., 60 (1985), 142-149. doi: 10.1016/0022-0396(85)90125-1. Google Scholar [39] P. Pucci and J. Serrin, Extensions of the mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210. doi: 10.1016/0022-1236(84)90072-7. Google Scholar [40] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. Math. Soc. Providence, RI, 1986. doi: 10.1090/cbms/065. Google Scholar [41] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410. doi: 10.1016/S0377-0427(99)00269-1. Google Scholar [42] R. Stegliński, On sequences of large solutions for discrete anisotropic equations, Electron. J. Qual. Theory Differ. Equ., 25 (2015), 1-10. Google Scholar [43] D. B. Wang and W. Guan, Three positive solutions of boundary value problems for pLaplacian difference equations, Comput. Math. Appl., 55 (2008), 1943-1949. doi: 10.1016/j.camwa.2007.08.033. Google Scholar [44] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, 1998. Google Scholar [45] P. J. Y. Wong and L. Xie, Three symmetric solutions of Lidstone boundary value problems for difference and partial difference equations, Comput. Math. Appl., 45 (2003), 1445-1460. doi: 10.1016/S0898-1221(03)00102-0. Google Scholar [46] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅱ/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0. Google Scholar
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