Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with \(\phi_c\)-Laplacian. (English) Zbl 1409.39004
Summary: In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the boundary value problems for a second-order \(\phi_c\)-Laplacian difference equation. To the best of our knowledge, this is the first time to discuss the existence of infinitely many positive solutions to the boundary value problems for difference equations involving the special \(\phi_c\)-Laplacian operator.
References:
| [1] | Mawhin, J., Periodic solutions of second order nonlinear difference systems with \(\phi \)-Laplacian: a variational approach, Nonlinear Anal., 75, 4672-4687 (2012) · Zbl 1252.39016 |
| [2] | Bonheure, D.; Habets, P.; Obersnel, F.; Omari, P., Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations, 243, 208-237 (2007) · Zbl 1136.34023 |
| [3] | Henderson, J.; Thompson, H. B., Existence of multiple solutions for second order discrete boundary value problems, Comput. Math. Appl., 43, 1239-1248 (2002) · Zbl 1005.39014 |
| [4] | Bereanu, C.; Mawhin, J., Boundary value problems for second-order nonlinear difference equations with discrete \(\phi \)-Laplacian and singular \(\phi \), J. Difference Equ. Appl., 14, 1099-1118 (2008) · Zbl 1161.39003 |
| [5] | Guo, Z. M.; Yu, J. S., The existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 46, 506-515 (2003) · Zbl 1215.39001 |
| [6] | Zhou, Z.; Yu, J. S.; Chen, Y. M., Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math., 54, 83-93 (2011) · Zbl 1239.39010 |
| [7] | Zhou, Z.; Yu, J. S., Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Appl. Sin. Engl. Ser., 29, 1809-1822 (2013) · Zbl 1284.39006 |
| [8] | Shi, H. P., Periodic and subharmonic solutions for second-order nonlinear difference equations, J. Appl. Math. Comput., 48, 157-171 (2015) · Zbl 1323.39013 |
| [9] | Zhou, Z.; Ma, D. F., Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math., 58, 781-790 (2015) · Zbl 1328.39011 |
| [10] | Lin, G. H.; Zhou, Z., Homoclinic solutions of discrete \(\phi \)-Laplacian equations with mixed nonlinearities, Comm. Pure Appl. Anal., 17, 1723-1747 (2018) · Zbl 1396.39008 |
| [11] | Tang, X. H., Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. (Engl. Ser.), 32, 463-473 (2016) · Zbl 1386.39015 |
| [12] | Zhang, Q. Q., Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143, 3155-3163 (2015) · Zbl 1351.37236 |
| [13] | Zhang, Q. Q., Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part, Comm. Pure Appl. Anal., 14, 1929-1940 (2017) · Zbl 1317.39011 |
| [14] | Zhou, Z.; Su, M. T., Boundary value problems for \(2 n\)-order \(\phi_c\)-Laplacian difference equations containing both advance and retardation, Appl. Math. Lett., 41, 7-11 (2015) · Zbl 1312.39013 |
| [15] | Bonanno, G.; Candito, P., Infinitely many solutions for a class of discrete non-linear boundary value problems, Appl. Anal., 88, 605-616 (2009) · Zbl 1176.39004 |
| [16] | Bonanno, G.; Jebelean, P.; Serban, C., Superlinear discrete problems, Appl. Math. Lett., 52, 162-168 (2016) · Zbl 1330.39001 |
| [17] | D’Agua, G.; Mawhin, J.; Sciammetta, A., Positive solutions for a discrete two point nonlinear boundary value problem with \(p\)-Laplacian, J. Math. Anal. Appl., 447, 383-397 (2017) · Zbl 1352.39002 |
| [18] | Ricceri, B., A general variational principle and some of its applications, J. Comput. Appl. Math., 133, 401-410 (2000) · Zbl 0946.49001 |
| [19] | Agarwal, R. P., Difference Equations and Inequalities. Theory, Methods, and Applications (2000), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York-Basel · Zbl 0952.39001 |
| [20] | Kelly, W. G.; Peterson, A. C., Difference Equations: An Introduction with Applications (1991), Academic Press: Academic Press San Diego, New York, Basel · Zbl 0733.39001 |
| [21] | Jiang, L.; Zhou, Z., Three solutions to Dirichlet boundary value problems for \(p\)-Laplacian difference equations, Adv. Diff. Equ., 2008, 1-10 (2008) · Zbl 1146.39028 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
