Periodic and Neumann problems for discrete \(p(\cdot )\)-Laplacian. (English) Zbl 1270.35243
Summary: Using critical point theory, we obtain the existence of solutions for some periodic boundary value problems involving the discrete \(p(\cdot )\)-Laplacian. These extend and improve known results for similar problems with discrete \(p\)-Laplacian. Similar results for Neumann problems are also provided. As applications we prove upper and lower solutions theorems for both considered cases.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
Keywords:
discrete \(p(\cdot )\)-Laplacian operator; critical point; Palais-Smale condition; lower and upper solutionsReferences:
| [1] | Agarwal, A.; Perera, K.; O’Regan, D., Multiple positive solutions of singular discrete \(p\)-Laplacian problems via variational methods, Adv. Difference Equ., 2005, 93-99 (2005) · Zbl 1098.39001 |
| [2] | Ahmad, S.; Lazer, A. C.; Paul, J. L., Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Math. J., 25, 933-944 (1976) · Zbl 0351.35036 |
| [3] | Bereanu, C.; Jebelean, P.; Şerban, C., Ground state and mountain pass solutions for discrete \(p(\cdot)\)-Laplacian, Bound. Value Probl., 2012, 104 (2012) · Zbl 1279.39003 |
| [4] | Bereanu, C.; Mawhin, J., Existence and multiplicity results for periodic solutions of nonlinear difference equations, J. Difference Equ. Appl., 12, 677-695 (2006) · Zbl 1103.39003 |
| [5] | Bereanu, C.; Thompson, H. B., Periodic solutions of nonlinear diference equations with discrete \(\phi \)-Laplacian, J. Math. Anal. Appl., 330, 1002-1015 (2007) · Zbl 1127.39006 |
| [6] | Cabada, A.; Iannizzotto, A.; Tersian, S., Multiple solutions for discrete boundary value problems, J. Math. Anal. Appl., 356, 418-428 (2009) · Zbl 1169.39008 |
| [7] | Galewski, M.; Gła̧b, S., On the discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl., 386, 956-965 (2012) · Zbl 1233.39004 |
| [9] | Guiro, A.; Nyanquini, I.; Ouaro, S., On the solvability of discrete nonlinear Neumann problems involving the \(p(x)\)-Laplacian, Adv. Difference Equ., 2011, 32 (2011) · Zbl 1291.39014 |
| [10] | Hamel, G., Ueber erzwungene Schingungen bei endlischen Amplituden, Math. Ann., 86, 1-13 (1922) · JFM 48.0519.03 |
| [11] | Jebelean, P.; Şerban, C., Ground state periodic solutions for difference equations with discrete \(p\)-Laplacian, Appl. Math. Comput., 217, 9820-9827 (2011) · Zbl 1228.39011 |
| [12] | Jiang, L.; Zhou, Z., Three solutions to Dirichlet boundary value problems for \(p\)-Laplacian difference equations, Adv. Difference Equ., 2008 (2008), Article ID 345916 · Zbl 1146.39028 |
| [13] | Koné, B.; Ouaro, S., Weak solutions for anisotropic discrete boundary value problems, J. Difference Equ. Appl., 16, 2, 1-11 (2010) |
| [14] | Koné, B.; Ouaro, S., On the solvability of discrete nonlinear two point boundary value problems, Int. J. Math. Math. Sci., 2012 (2012), Article ID 927607 · Zbl 1253.39007 |
| [15] | Mashiyev, R.; Yucedag, Z.; Ogras, S., Existence and multiplicity of solutions for a Dirichlet problem involving the discrete \(p(x)\)-Laplacian operator, Electron. J. Qual. Theory Differ. Equ., 67, 1-10 (2011) · Zbl 1340.47038 |
| [16] | Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelberg · Zbl 0676.58017 |
| [17] | Mihăilescu, M.; Rădulescu, V.; Tersian, S., Eigenvalue problems for anisotropic discrete boundary value problems, J. Difference Equ. Appl., 15, 6, 557-567 (2009) · Zbl 1181.47016 |
| [18] | Rabinowitz, P. H., (Minimax Methods in Critical Point Theory with Applications to Differential Equations. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conferences in Mathematics, vol. 65 (1986), American Mathematical Society: American Mathematical Society Providence/Rhode Island) · Zbl 0609.58002 |
| [19] | Şerban, C., Existence of solutions for discrete \(p\)-Laplacian with potential boundary conditions, J. Difference Equ. Appl. (2012) |
| [20] | Tian, Y.; Du, Z.; Ge, W., Existence results for discrete Sturm-Liouville problem via variational methods, J. Difference Equ. Appl., 13, 6, 467-478 (2007) · Zbl 1129.39007 |
| [21] | Tian, Y.; Ge, W., The existence of solutions for a second-order discrete Neumann problem with a \(p\)-Laplacian, J. Appl. Math. Comput., 26, 333-340 (2008) · Zbl 1156.39009 |
| [22] | Yang, Y.; Zhang, J., Existence of solutions for some discrete value problems with a parameter, Appl. Math. Comput., 211, 293-302 (2009) · Zbl 1169.39009 |
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