Existence of solutions for a nonlinear discrete system involving the \(p\)-Laplacian. (English) Zbl 1249.39009

The authors consider the discrete boundary value problem \[ \Delta \bigl (\Phi _p(\Delta u(t-1))\bigr )+\lambda \nabla F(t,u(t))=0,\quad t\in [1,M]_{\mathbb Z},\; u(0)=0=u(M+1), \] where \(\Phi _p(s)=| s| ^{p-1}s\), \(\lambda >0\) is a parameter and \(F\:[0,M]_{\mathbb Z}\times {\mathbb R}^m\to {\mathbb R}\) is continuously differentiable. Using the critical point theory, various existence statements for solvability of the above problem are proved. A typical result is the following theorem:
Suppose that \(F(0,0)=0\) and there exists a number \(\beta >p\) such that \[ \limsup _{| x| \to \infty } \frac {F(t,x)}{| x| ^{\beta }}>0\quad \text{for all }t\in [0,M]_{\mathbb Z}. \] If \(\lambda >0\), then the above problem has at least one solution.


39A12 Discrete version of topics in analysis
39A10 Additive difference equations
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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[1] D. Bai, Y. Xu: Nontrivial solutions of boundary value problems of second-order difference equations. J. Math. Anal. Appl. 326 (2007), 297–302. · Zbl 1113.39018
[2] G. Bonanno, P. Candito: Nonlinear difference equations investigated via critical points methods. Nonlinear Anal., Theory Methods Appl. 70 (2009), 3180–3186. · Zbl 1166.39006
[3] G. Bonanno, P. Candito: Infinitely many solutions for a class of discrete non-linear boundary value problems. Appl. Anal. 88 (2009), 605–616. · Zbl 1176.39004
[4] P. Candito, N. Giovannelli: Multiple solutions for a discrete boundary value problem involving the p-Laplacian. Comput. Math. Appl. 56 (2008), 959–964. · Zbl 1155.39301
[5] Z.M. Guo, J. S. Yu: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci. China Ser. A 46 (2003), 506–515. · Zbl 1215.39001
[6] Z.M. Guo, J. S. Yu: The existence of periodic and subharmonic solutions to subquadratic second order difference equations. J. Lond. Math. Soc., II. Ser. 68 (2003), 419–430. · Zbl 1046.39005
[7] J. Kuang: Applied Inequalities. Shandong Science and Technology Press, Jinan City, 2004. (In Chinese.)
[8] W.D. Lu: Variational Methods in Differential Equations. Scientific Publishing House in China, 2002.
[9] J. Ma, C. L. Tang: Periodic solutions for some nonautonomous second order systems. J. Math. Anal. Appl. 275 (2002), 482–494. · Zbl 1024.34036
[10] J. Mawhin, M. Willem: Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York, 1989. · Zbl 0676.58017
[11] P.H. Rabinowitz: Minimax Methods in Critical Point Theory with Application to Differential Equations. Reg. Conf. Ser. Math, 65. Am. Math. Soc., Provindence, 1986. · Zbl 0609.58002
[12] C.-L. Tang, X.-P. Wu: Notes on periodic solutions of subquadratic second order systems. J. Math. Anal. Appl. 285 (2003), 8–16. · Zbl 1054.34075
[13] J.F. Wu, X.P. Wu: Existence of nontrivial periodic solutions for a class of superquadratic second-order Hamiltonian systems. J. Southwest Univ. (Natural Science Edition) 30 (2008), 26–31.
[14] X.-P. Wu, C.-L. Tang: Periodic solution of a class of non-autonomous second order systems. J. Math. Anal. Appl. 236 (1999), 227–235. · Zbl 0971.34027
[15] Y.-F. Xue, C.-L. Tang: Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems. Appl. Math. Comput. 196 (2008), 494–500. · Zbl 1153.39024
[16] Y.-F. Xue, C.-L. Tang: Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system. Nonlinear Anal., Theory Methods Appl. 67 (2007), 2072–2080. · Zbl 1129.39008
[17] X. Zhang, X. Tang: Existence of nontrivial solutions for boundary value problems of second-order discrete systems. Math. Slovaca 61 (2011), 769–778. · Zbl 1274.39018
[18] F. Zhao, X. Wu: Periodic solutions for a class of nonautonomous second order systems. J. Math. Anal. Appl. 296 (2004), 422–434. · Zbl 1050.34062
[19] Z. Zhou, J.-S. Yu, Z.-M. Guo: Periodic solutions of higher-dimensional discrete systems. Proc. R. Soc. Edinb., Sect. A, Math. 134 (2004), 1013–1022. · Zbl 1073.39010
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