`a`
Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Multiplicity results for discrete anisotropic equations
Pages: 203 - 218, Issue 1, January 2018

doi:10.3934/dcdsb.2018014      Abstract        References        Full text (450.2K)           Related Articles

Marek Galewski - Institute of Mathematics, Technical University of Lodz, Wolczanska 215, 90-924 Lodz, Poland (email)
Shapour Heidarkhani - Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran (email)
Amjad Salari - Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran (email)

1 A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.       
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.       
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.       
4 M. Bendahmane and K. H. Karlsen, Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$-data, Commun. Pure Appl. Anal., 5 (2006), 733-762.       
5 M. Bendahmane, M. Langlais and M. Saad, On some anisotropic reaction-diffusion systems with $L^1$-data modeling the propagation of an epidemic disease, Nonlinear Anal. TMA, 54 (2003), 617-636.       
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.       
7 G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. TMA, 75 (2012), 2992-3007.       
8 G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal. TMA, 70 (2009), 3180-3186.       
9 G. Bonanno, P. Jebelean and C. Serban, Three solutions for discrete anisotropic periodic and Neumann problems, Dynamic Sys. Appl., 22 (2013), 183-196.       
10 A. Cabada, A. Iannizzotto and S. Tersian, Multiple solutions for discrete boundary value problem, J. Math. Anal. Appl., 356 (2009), 418-428.       
11 P. Candito and N. Giovannelli, Multiple solutions for a discrete boundary value problem, Comput. Math. Appl., 56 (2008), 959-964.       
12 J. Chu and D. Jiang, Eigenvalues and discrete boundary value problems for the one-dimensional $p$-Laplacian, J. Math. Anal. Appl., 305 (2005), 452-465.       
13 G. D'Aguì, Multiplicity results for nonlinear mixed boundary value problem, Bound. Value Probl., 2012 (2012), 1-12.       
14 E. Eisenriegler, Anisotropic colloidal particles in critical fluids, J. Chem. Phys., 121 (2004), p3299.
15 A. El Hamidi and J. Vétois, Sharp Sobolev asymptotics for critical anisotropic equations, Arch. Ration. Mech. Anal., 192 (2009), 1-36.       
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.       
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.       
19 M. Galewski, S. 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.       
21 M. Galewski and R. Wieteska, On the system of anisotropic discrete BVPs, J. Differ. Equ. Appl., 19 (2013), 1065-1081.       
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.       
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.       
24 S. Heidarkhani and M. Khaleghi Moghadam, Existence of three solutions for perturbed nonlinear difference equations, Opuscula Math., 34 (2014), 747-761.       
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.       
26 S. Heidarkhani and A. Salari, Nontrivial solutions for impulsive fractional differential systems through variational methods, Comput. Math. Appl., (2016).
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 Moghadam, S. Heidarkhani and J. Henderson, Infinitely many solutions for perturbed difference equations, J. Differ. Equ. Appl., 20 (2014), 1055-1068.       
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.       
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.       
32 P. Lindqvist, On the equation div$(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0,$ Proc. Amer. Math. Soc., 109 (1990), 157-164.       
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.       
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.       
35 M. Mihailescu, V. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems, J. Differ. Equ. Appl., 15 (2009), 557-567.       
36 G. Molica Bisci and D. Repovš, Existence of solutions for $p$-Laplacian discrete equations, Appl. Math. Comput., 242 (2014), 454-461.       
37 G. Molica Bisci and D. Repovš, On sequences of solutions for discrete anisotropic equations, Expo. Math., 32 (2014), 284-295.       
38 P. Pucci and J. Serrin, A mountain pass theorem, J. Differ. Eqs., 60 (1985), 142-149.       
39 P. Pucci and J. Serrin, Extensions of the mountain pass theorem, J. Funct. Anal., 59 (1984), 185-210.       
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.       
41 B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410.       
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 $p$-Laplacian difference equations, Comput. Math. Appl., 55 (2008), 1943-1949.       
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.       
46 E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B, Springer-Verlag, New York, 1990.       

Go to top