Multiple periodic solutions to a discrete p^{(k)} - Laplacian problem

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October 2014, 19(8): 2535-2547. doi: 10.3934/dcdsb.2014.19.2535

Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem

Marek Galewski ^1, and Renata Wieteska ^1,

1.	Institute of Mathematics, Technical University of Lodz, Wolczanska 215, 90-924 Lodz, Poland

Received November 2013 Revised November 2013 Published August 2014

Abstract
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We investigate the existence of multiple periodic solutions to the anisotropic discrete system. We apply the linking method and a new three critical point theorem which we provide.

Keywords: non-smooth problems, Discrete boundary value problem, local linking., three critical point theorem.

Mathematics Subject Classification: Primary: 39A10; Secondary: 34B15, 65Q2.

Citation: Marek Galewski, Renata Wieteska. Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2535-2547. doi: 10.3934/dcdsb.2014.19.2535

References:

[1]	R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992.
[2]	C. Bereanu, P. Jebelean and C. Şerban, Periodic and Neumann problems for discrete $p(\cdot )-$Laplacian, J. Math. Anal. Appl., 399 (2013), 75-87. doi: 10.1016/j.jmaa.2012.09.047.
[3]	C. Bereanu, P. Jebelean and C. Şerban, Ground state and mountain pass solutions for discrete $p(\cdot )-$Laplacian, Bound. Value Probl., 104 (2012), 1-13. doi: 10.1186/1687-2770-2012-104.
[4]	G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal., Theory Methods Appl. A, 70 (2009), 3180-3186. doi: 10.1016/j.na.2008.04.021.
[5]	G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differ. Eqs, 244 (2008), 3031-3059. doi: 10.1016/j.jde.2008.02.025.
[6]	G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1-10. doi: 10.1080/00036810903397438.
[7]	A. Cabada and S. Tersian, Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Appl. Math. Comput., 219 (2013), 5261-5267. doi: 10.1016/j.amc.2012.11.066.
[8]	Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522.
[9]	S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1.
[10]	X. L. Fan and H. Zhang, Existence of solutions for $p(x)-$Lapacian dirichlet problem, Nonlinear Anal., Theory Methods Appl., 52 (2003), 1843-1852. doi: 10.1016/S0362-546X(02)00150-5.
[11]	M. Galewski and R. Wieteska, A note on the multiplicity of solutions to anisotropic discrete BVP's, Appl. Math. Lett., 26 (2013), 524-529. doi: 10.1016/j.aml.2012.11.002.
[12]	A. Guiro, I. Nyanquini and S. Ouaro, On the solvability of discrete nonlinear Neumann problems involving the $p(x)-$Laplacian, Adv. Difference Equ., 32 (2011), 14 pp.
[13]	P. Harjulehto, P. Hästö, U. V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574. doi: 10.1016/j.na.2010.02.033.
[14]	B. Kone and S. Ouaro, Weak solutions for anisotropic discrete boundary value problems, J. Difference Equ. Appl., 17 (2011), 1537-1547. doi: 10.1080/10236191003657246.
[15]	V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Academic Press, New York, 1988.
[16]	S. Liu, Multiple periodic solutions for nonlinear difference systems involving the p-Laplacian, J. Difference Equ. Appl., 17 (2011), 1591-1598. doi: 10.1080/10236191003730480.
[17]	N. Marcu and G. Molica Bisci, Existence and multiplicity results for nonlinear discrete inclusions, Electron. J. Differential Equations, (2012), 1-13.
[18]	M. Mihăilescu, V. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems. J. Difference Equ. Appl., 15 (2009), 557-567. doi: 10.1080/10236190802214977.
[19]	G. Molica Bisci and D. Repovs, On some variational algebraic problems, Adv. Nonlinear Analysis, 2 (2013), 127-146.
[20]	G. Molica Bisci and D. Repovs, Nonlinear Algebraic Systems with discontinuous terms, J. Math. Anal. Appl., 398 (2013), 846-856. doi: 10.1016/j.jmaa.2012.09.046.
[21]	M. Růžička, Electrorheological Fluids: Modelling and Mathematical Theory, Lecture Notes in Mathematics, Vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.
[22]	P. Stehlík, On variational methods for periodic discrete problems, J. Difference Equ. Appl., 14 (2008), 259-273. doi: 10.1080/10236190701483160.
[23]	Y. Tian, Z. Du and W. Ge, Existence results for discrete Sturm-Liouville problem via variational methods, J. Difference Equ. Appl., 13 (2007), 467-478. doi: 10.1080/10236190601086451.
[24]	M. Willem, Minimax Theorem, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.
[25]	V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.

show all references

References:

[1]	R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992.
[2]	C. Bereanu, P. Jebelean and C. Şerban, Periodic and Neumann problems for discrete $p(\cdot )-$Laplacian, J. Math. Anal. Appl., 399 (2013), 75-87. doi: 10.1016/j.jmaa.2012.09.047.
[3]	C. Bereanu, P. Jebelean and C. Şerban, Ground state and mountain pass solutions for discrete $p(\cdot )-$Laplacian, Bound. Value Probl., 104 (2012), 1-13. doi: 10.1186/1687-2770-2012-104.
[4]	G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal., Theory Methods Appl. A, 70 (2009), 3180-3186. doi: 10.1016/j.na.2008.04.021.
[5]	G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differ. Eqs, 244 (2008), 3031-3059. doi: 10.1016/j.jde.2008.02.025.
[6]	G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1-10. doi: 10.1080/00036810903397438.
[7]	A. Cabada and S. Tersian, Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Appl. Math. Comput., 219 (2013), 5261-5267. doi: 10.1016/j.amc.2012.11.066.
[8]	Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522.
[9]	S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1.
[10]	X. L. Fan and H. Zhang, Existence of solutions for $p(x)-$Lapacian dirichlet problem, Nonlinear Anal., Theory Methods Appl., 52 (2003), 1843-1852. doi: 10.1016/S0362-546X(02)00150-5.
[11]	M. Galewski and R. Wieteska, A note on the multiplicity of solutions to anisotropic discrete BVP's, Appl. Math. Lett., 26 (2013), 524-529. doi: 10.1016/j.aml.2012.11.002.
[12]	A. Guiro, I. Nyanquini and S. Ouaro, On the solvability of discrete nonlinear Neumann problems involving the $p(x)-$Laplacian, Adv. Difference Equ., 32 (2011), 14 pp.
[13]	P. Harjulehto, P. Hästö, U. V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574. doi: 10.1016/j.na.2010.02.033.
[14]	B. Kone and S. Ouaro, Weak solutions for anisotropic discrete boundary value problems, J. Difference Equ. Appl., 17 (2011), 1537-1547. doi: 10.1080/10236191003657246.
[15]	V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Academic Press, New York, 1988.
[16]	S. Liu, Multiple periodic solutions for nonlinear difference systems involving the p-Laplacian, J. Difference Equ. Appl., 17 (2011), 1591-1598. doi: 10.1080/10236191003730480.
[17]	N. Marcu and G. Molica Bisci, Existence and multiplicity results for nonlinear discrete inclusions, Electron. J. Differential Equations, (2012), 1-13.
[18]	M. Mihăilescu, V. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems. J. Difference Equ. Appl., 15 (2009), 557-567. doi: 10.1080/10236190802214977.
[19]	G. Molica Bisci and D. Repovs, On some variational algebraic problems, Adv. Nonlinear Analysis, 2 (2013), 127-146.
[20]	G. Molica Bisci and D. Repovs, Nonlinear Algebraic Systems with discontinuous terms, J. Math. Anal. Appl., 398 (2013), 846-856. doi: 10.1016/j.jmaa.2012.09.046.
[21]	M. Růžička, Electrorheological Fluids: Modelling and Mathematical Theory, Lecture Notes in Mathematics, Vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.
[22]	P. Stehlík, On variational methods for periodic discrete problems, J. Difference Equ. Appl., 14 (2008), 259-273. doi: 10.1080/10236190701483160.
[23]	Y. Tian, Z. Du and W. Ge, Existence results for discrete Sturm-Liouville problem via variational methods, J. Difference Equ. Appl., 13 (2007), 467-478. doi: 10.1080/10236190601086451.
[24]	M. Willem, Minimax Theorem, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.
[25]	V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.

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