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Positive solutions of the discrete Robin problem with $\phi$-Laplacian

 1 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China 2 Center for Applied Mathematics Guangzhou University Guangzhou 510006, China

* Corresponding author: Zhan Zhou

Received  April 2019 Revised  October 2019 Published  April 2020

In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Robin problem with $\phi$-Laplacian. We show that, an unbounded sequence of positive solutions and a sequence of positive solutions which converges to zero will emerge from the suitable oscillating behavior of the nonlinear term at infinity and at the zero, respectively. We also give two examples to illustrate our main results.

Citation: Jiaoxiu Ling, Zhan Zhou. Positive solutions of the discrete Robin problem with $\phi$-Laplacian. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020338
References:
 [1] R. P. Agarwal, D. O'Regan and J. Y. P. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-015-9171-3.  Google Scholar [2] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Second edition, Monographs and Textbooks in Pure and Applied Mathematics, 228. Marcel Dekker, Inc., New York, 2000.  Google Scholar [3] Z. Balanov, C. Garcia-Azpeitia and W. Krawcewicz, On variational and topological methods in nonlinear difference equations, Communications on Pure and Applied Analysis, 17 (2018), 2813-2844.  doi: 10.3934/cpaa.2018133.  Google Scholar [4] C. Bereanu and J. Mawhin, Boundary value problems for second-order nonlinear difference equations with discrete $\phi$-Laplacian and singular $\phi$, Journal of Difference Equations and Applications, 14 (2008), 1099-1118.  doi: 10.1080/10236190802332290.  Google Scholar [5] G. Bonanno and P. Candito, Infinitely many solutions for a class of discrete non-linear boundary value problems, Applicable Analysis, 88 (2009), 605-616.  doi: 10.1080/00036810902942242.  Google Scholar [6] G. Bonanno, R. Livrea and J. Mawhin, Existence results for parametric boundary value problems involving the mean curvature operator, Nonlinear Differential Equations and Applications, 22 (2015), 411-426.  doi: 10.1007/s00030-014-0289-7.  Google Scholar [7] G. Bonanno, P. Jebelean and C. Şerban, Superlinear discrete problems, Applied Mathematics Letters, 52 (2016), 162-168.  doi: 10.1016/j.aml.2015.09.005.  Google Scholar [8] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, Journal of Differential Equations, 243 (2007), 208-237.  doi: 10.1016/j.jde.2007.05.031.  Google Scholar [9] G. D'Aguìa, J. Mawhin and A. Sciammetta, Positive solutions for a discrete two point nonlinear boundary value problem with $p$-Laplacian, Journal of Mathematical Analysis and Applications, 447 (2017), 383-397.  doi: 10.1016/j.jmaa.2016.10.023.  Google Scholar [10] L. Erbe, B. G. Jia and Q. Q. Zhang, Homoclinic solutions of discrete nonlinear systems via variational method, Journal of Applied Analysis and Computation, 9 (2019), 271-294.  doi: 10.11948/2019.271.  Google Scholar [11] Z. M. Guo and J. S. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 46 (2003), 506-515.  doi: 10.1007/BF02884022.  Google Scholar [12] J. Henderson and H. B. Thompson, Existence of multiple solutions for second order discrete boundary value problems, Computers and Mathematics with Applications, 43 (2002), 1239-1248.  doi: 10.1016/S0898-1221(02)00095-0.  Google Scholar [13] G. H. Lin and Z. Zhou, Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities, Communications on Pure and Applied Analysis, 17 (2018), 1723-1747.  doi: 10.3934/cpaa.2018082.  Google Scholar [14] G. H. Lin, Z. Zhou and J. S. Yu, Ground state solutions of discrete asymptotically linear Schrödinge equations with bounded and non-periodic potentials, Journal of Dynamics and Differential Equations, (2019). https://doi.org/10.1007/s10884-019-09743-4. Google Scholar [15] J. X. Ling and Z. Zhou, Positive solutions of the discrete Dirichlet problem involving the mean curvature operator, Open Mathematics, 17 (2019), 1055-1064.  doi: 10.1515/math-2019-0081.  Google Scholar [16] Y. H. Long and B. L. Zeng, Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence, Open Mathematics, 15 (2017), 1549-1557.  doi: 10.1515/math-2017-0129.  Google Scholar [17] W. G. Kelly and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, Inc., Boston, MA, 1991.   Google Scholar [18] J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: A variational approach, Nonlinear Analysis, 75 (2012), 4672-4687.  doi: 10.1016/j.na.2011.11.018.  Google Scholar [19] F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, Journal of Differential Equations, 249 (2010), 1674-1725.  doi: 10.1016/j.jde.2010.07.001.  Google Scholar [20] B. Ricceri, A general variational principle and some of its applications, Journal of Computational and Applied Mathematics, 133 (2000), 401-410.  doi: 10.1016/S0377-0427(99)00269-1.  Google Scholar [21] H. P. Shi, Periodic and subharmonic solutions for second-order nonlinear difference equations, Journal of Applied Mathematics and Computing, 48 (2015), 157-171.  doi: 10.1007/s12190-014-0796-z.  Google Scholar [22] X. H. Tang, Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. (Engl. Ser.), 32 (2016), 463-473.  doi: 10.1007/s10114-016-4262-8.  Google Scholar [23] J. S. Yu and B. Zheng, Modeling Wolbachia infection in mosquito population via discrete dynamical model, Journal of Difference Equations and Applications, 25 (2019), 1549-1567.  doi: 10.1080/10236198.2019.1669578.  Google Scholar [24] Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proceedings of the American Mathematical Society, 143 (2015), 3155-3163.  doi: 10.1090/S0002-9939-2015-12107-7.  Google Scholar [25] Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part, Communications on Pure and Applied Analysis, 14 (2015), 1929-1940.  doi: 10.3934/cpaa.2015.14.1929.  Google Scholar [26] Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions, Communications on Pure and Applied Analysis, 18 (2019), 425-434.  doi: 10.3934/cpaa.2019021.  Google Scholar [27] Z. Zhou, J. S. Yu and Y. M. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Science China Mathematics, 54 (2011), 83-93.  doi: 10.1007/s11425-010-4101-9.  Google Scholar [28] Z. Zhou and J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Sin. (Engl. Ser.), 29 (2013), 1809-1822.  doi: 10.1007/s10114-013-0736-0.  Google Scholar [29] Z. Zhou and D. F. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Science China Mathematics, 58 (2015), 781-790.  doi: 10.1007/s11425-014-4883-2.  Google Scholar [30] Z. Zhou and M. T. Su, Boundary value problems for 2$n$th-order $\phi_{c}$-Laplacian difference equations containing both advance and retardation, Applied Mathematics Letters, 41 (2015), 7-11. doi: 10.1016/j.aml.2014.10.006.  Google Scholar [31] Z. Zhou and J. X. Ling, Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with $\phi_c$-Laplacian, Applied Mathematics Letters, 91 (2019), 28-34.  doi: 10.1016/j.aml.2018.11.016.  Google Scholar

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References:
 [1] R. P. Agarwal, D. O'Regan and J. Y. P. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-015-9171-3.  Google Scholar [2] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Second edition, Monographs and Textbooks in Pure and Applied Mathematics, 228. Marcel Dekker, Inc., New York, 2000.  Google Scholar [3] Z. Balanov, C. Garcia-Azpeitia and W. Krawcewicz, On variational and topological methods in nonlinear difference equations, Communications on Pure and Applied Analysis, 17 (2018), 2813-2844.  doi: 10.3934/cpaa.2018133.  Google Scholar [4] C. Bereanu and J. Mawhin, Boundary value problems for second-order nonlinear difference equations with discrete $\phi$-Laplacian and singular $\phi$, Journal of Difference Equations and Applications, 14 (2008), 1099-1118.  doi: 10.1080/10236190802332290.  Google Scholar [5] G. Bonanno and P. Candito, Infinitely many solutions for a class of discrete non-linear boundary value problems, Applicable Analysis, 88 (2009), 605-616.  doi: 10.1080/00036810902942242.  Google Scholar [6] G. Bonanno, R. Livrea and J. Mawhin, Existence results for parametric boundary value problems involving the mean curvature operator, Nonlinear Differential Equations and Applications, 22 (2015), 411-426.  doi: 10.1007/s00030-014-0289-7.  Google Scholar [7] G. Bonanno, P. Jebelean and C. Şerban, Superlinear discrete problems, Applied Mathematics Letters, 52 (2016), 162-168.  doi: 10.1016/j.aml.2015.09.005.  Google Scholar [8] D. Bonheure, P. Habets, F. Obersnel and P. Omari, Classical and non-classical solutions of a prescribed curvature equation, Journal of Differential Equations, 243 (2007), 208-237.  doi: 10.1016/j.jde.2007.05.031.  Google Scholar [9] G. D'Aguìa, J. Mawhin and A. Sciammetta, Positive solutions for a discrete two point nonlinear boundary value problem with $p$-Laplacian, Journal of Mathematical Analysis and Applications, 447 (2017), 383-397.  doi: 10.1016/j.jmaa.2016.10.023.  Google Scholar [10] L. Erbe, B. G. Jia and Q. Q. Zhang, Homoclinic solutions of discrete nonlinear systems via variational method, Journal of Applied Analysis and Computation, 9 (2019), 271-294.  doi: 10.11948/2019.271.  Google Scholar [11] Z. M. Guo and J. S. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 46 (2003), 506-515.  doi: 10.1007/BF02884022.  Google Scholar [12] J. Henderson and H. B. Thompson, Existence of multiple solutions for second order discrete boundary value problems, Computers and Mathematics with Applications, 43 (2002), 1239-1248.  doi: 10.1016/S0898-1221(02)00095-0.  Google Scholar [13] G. H. Lin and Z. Zhou, Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities, Communications on Pure and Applied Analysis, 17 (2018), 1723-1747.  doi: 10.3934/cpaa.2018082.  Google Scholar [14] G. H. Lin, Z. Zhou and J. S. Yu, Ground state solutions of discrete asymptotically linear Schrödinge equations with bounded and non-periodic potentials, Journal of Dynamics and Differential Equations, (2019). https://doi.org/10.1007/s10884-019-09743-4. Google Scholar [15] J. X. Ling and Z. Zhou, Positive solutions of the discrete Dirichlet problem involving the mean curvature operator, Open Mathematics, 17 (2019), 1055-1064.  doi: 10.1515/math-2019-0081.  Google Scholar [16] Y. H. Long and B. L. Zeng, Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence, Open Mathematics, 15 (2017), 1549-1557.  doi: 10.1515/math-2017-0129.  Google Scholar [17] W. G. Kelly and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, Inc., Boston, MA, 1991.   Google Scholar [18] J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: A variational approach, Nonlinear Analysis, 75 (2012), 4672-4687.  doi: 10.1016/j.na.2011.11.018.  Google Scholar [19] F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation, Journal of Differential Equations, 249 (2010), 1674-1725.  doi: 10.1016/j.jde.2010.07.001.  Google Scholar [20] B. Ricceri, A general variational principle and some of its applications, Journal of Computational and Applied Mathematics, 133 (2000), 401-410.  doi: 10.1016/S0377-0427(99)00269-1.  Google Scholar [21] H. P. Shi, Periodic and subharmonic solutions for second-order nonlinear difference equations, Journal of Applied Mathematics and Computing, 48 (2015), 157-171.  doi: 10.1007/s12190-014-0796-z.  Google Scholar [22] X. H. Tang, Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. (Engl. Ser.), 32 (2016), 463-473.  doi: 10.1007/s10114-016-4262-8.  Google Scholar [23] J. S. Yu and B. Zheng, Modeling Wolbachia infection in mosquito population via discrete dynamical model, Journal of Difference Equations and Applications, 25 (2019), 1549-1567.  doi: 10.1080/10236198.2019.1669578.  Google Scholar [24] Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proceedings of the American Mathematical Society, 143 (2015), 3155-3163.  doi: 10.1090/S0002-9939-2015-12107-7.  Google Scholar [25] Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part, Communications on Pure and Applied Analysis, 14 (2015), 1929-1940.  doi: 10.3934/cpaa.2015.14.1929.  Google Scholar [26] Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions, Communications on Pure and Applied Analysis, 18 (2019), 425-434.  doi: 10.3934/cpaa.2019021.  Google Scholar [27] Z. Zhou, J. S. Yu and Y. M. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity, Science China Mathematics, 54 (2011), 83-93.  doi: 10.1007/s11425-010-4101-9.  Google Scholar [28] Z. Zhou and J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Sin. (Engl. Ser.), 29 (2013), 1809-1822.  doi: 10.1007/s10114-013-0736-0.  Google Scholar [29] Z. Zhou and D. F. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Science China Mathematics, 58 (2015), 781-790.  doi: 10.1007/s11425-014-4883-2.  Google Scholar [30] Z. Zhou and M. T. Su, Boundary value problems for 2$n$th-order $\phi_{c}$-Laplacian difference equations containing both advance and retardation, Applied Mathematics Letters, 41 (2015), 7-11. doi: 10.1016/j.aml.2014.10.006.  Google Scholar [31] Z. Zhou and J. X. Ling, Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with $\phi_c$-Laplacian, Applied Mathematics Letters, 91 (2019), 28-34.  doi: 10.1016/j.aml.2018.11.016.  Google Scholar
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