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 and 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.

[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.

[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.

[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.

[5]

M. BendahmaneM. 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.

[6]

M. BojowaldH. 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.

[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.

[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.

[9]

G. BonannoP. Jebelean and C. Serban, Three solutions for discrete anisotropic periodic and Neumann problems, Dynamic Sys. Appl., 22 (2013), 183-196. 

[10]

A. CabadaA. 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.

[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.

[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.

[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.

[14]

E. Eisenriegler, Anisotropic colloidal particles in critical fluids, J. Chem. Phys. 121 (2004), p3299. doi: 10.1063/1.1768514.

[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.

[16]

I. FragalaF. 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.

[17]

H. Gajewski, K. Groeger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1974.

[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.

[19]

M. GalewskiS. Głąb and R. Wieteska, Positive solutions for anisotropic discrete boundary value problems, Electron. J. Differ. Equ., 2013 (2013), 1-9. 

[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.

[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.

[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.

[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.

[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.

[25]

S. HeidarkhaniM. FerraraA. 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.

[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.

[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.

[28]

W. G. Kelly and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, New York, Basel, 1991.

[29]

M. Khaleghi MoghadamS. 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.

[30]

A. KristályM. 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.

[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.

[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.

[33]

M. MihailescuP. 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.

[34]

M. MihailescuP. 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.

[35]

M. MihailescuV. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems, J. Differ. Equ. Appl., 15 (2009), 557-567.  doi: 10.1080/10236190802214977.

[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.

[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.

[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.

[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.

[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.

[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.

[42]

R. Stegliński, On sequences of large solutions for discrete anisotropic equations, Electron. J. Qual. Theory Differ. Equ., 25 (2015), 1-10. 

[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.

[44]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, 1998.

[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.

[46]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅱ/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

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.

[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.

[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.

[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.

[5]

M. BendahmaneM. 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.

[6]

M. BojowaldH. 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.

[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.

[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.

[9]

G. BonannoP. Jebelean and C. Serban, Three solutions for discrete anisotropic periodic and Neumann problems, Dynamic Sys. Appl., 22 (2013), 183-196. 

[10]

A. CabadaA. 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.

[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.

[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.

[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.

[14]

E. Eisenriegler, Anisotropic colloidal particles in critical fluids, J. Chem. Phys. 121 (2004), p3299. doi: 10.1063/1.1768514.

[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.

[16]

I. FragalaF. 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.

[17]

H. Gajewski, K. Groeger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1974.

[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.

[19]

M. GalewskiS. Głąb and R. Wieteska, Positive solutions for anisotropic discrete boundary value problems, Electron. J. Differ. Equ., 2013 (2013), 1-9. 

[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.

[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.

[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.

[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.

[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.

[25]

S. HeidarkhaniM. FerraraA. 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.

[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.

[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.

[28]

W. G. Kelly and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, New York, Basel, 1991.

[29]

M. Khaleghi MoghadamS. 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.

[30]

A. KristályM. 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.

[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.

[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.

[33]

M. MihailescuP. 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.

[34]

M. MihailescuP. 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.

[35]

M. MihailescuV. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems, J. Differ. Equ. Appl., 15 (2009), 557-567.  doi: 10.1080/10236190802214977.

[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.

[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.

[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.

[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.

[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.

[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.

[42]

R. Stegliński, On sequences of large solutions for discrete anisotropic equations, Electron. J. Qual. Theory Differ. Equ., 25 (2015), 1-10. 

[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.

[44]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner-Verlag, Stuttgart, 1998.

[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.

[46]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅱ/B, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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