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MULTIPLE SOLUTIONS TO DISCRETE BOUNDARY VALUE PROBLEMS FOR THE p-LAPLACIAN WITH POTENTIAL TERMS ON FINITE GRAPHS

  • CHUNG, SOON-YEONG (DEPARTMENT OF MATHEMATICS AND PROGRAM OF INTEGRATED BIOTECHNOLOGY SOGANG UNIVERSITY) ;
  • PARK, JEA-HYUN (DEPARTMENT OF MATHEMATICS KUNSAN NATIONAL UNIVERSITY)
  • Received : 2014.08.11
  • Published : 2015.09.30

Abstract

In this paper, we prove the existence of at least three nontrivial solutions to nonlinear discrete boundary value problems $$\{^{-{\Delta}_{p,{\omega}}u(x)+V(x){\mid}u(x){\mid}^{q-2}u(x)=f(x,u(x)),x{\in}S,}_{u(x)=0,\;x{\in}{\partial}S}$$, involving the discrete p-Laplacian on simple, nite and connected graphs $\bar{S}(S{\cup}{\partial}S,E)$ with weight ${\omega}$, where 1 < q < p < ${\infty}$. The approach is based on a suitable combine of variational and truncations methods.

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

References

  1. R. P. Agarwal, K. Perera, and D. O'Regan, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (2004), no. 1-2, 69-73. https://doi.org/10.1016/j.na.2003.11.012
  2. R. P. Agarwal, K. Perera, and D. O'Regan, Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Difference Equ. 2005 (2005), no. 2, 93-99.
  3. G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal. 70 (2009), no. 9, 3180-3186. https://doi.org/10.1016/j.na.2008.04.021
  4. G. Bonanno, P. Candito, and G. D'Agui, Variational methods on finite dimensional Banach spaces and discrete problems, Adv. Nonlinear Stud. 14 (2014), no. 4, 915-39. https://doi.org/10.1515/ans-2014-0406
  5. P. Candito and N. Giovannelli, Multiple solutions for a discrete boundary value problem involving the p-Laplacian, Comput. Math. Appl. 56 (2008), no. 4, 959-964. https://doi.org/10.1016/j.camwa.2008.01.025
  6. A. Elmoataz, O. Lezoray, and S. Bougleux, Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing, IEEE Trans. Image Process. 17 (2008), no. 7, 1047-1060. https://doi.org/10.1109/TIP.2008.924284
  7. L. Gao, Existence of multiple solutions for a second-order difference equation with a parameter, Appl. Math. Comput. 216 (2010), no. 5, 1592-1598. https://doi.org/10.1016/j.amc.2010.03.012
  8. S.-Y. Ha, K. Lee, and D. Levy, Emergence of time-asymptotic ocking in a stochastic Cucker-Smale system, Commun. Math. Sci. 7 (2009), no. 2, 453-469. https://doi.org/10.4310/CMS.2009.v7.n2.a9
  9. S.-Y. Ha and D. Levy, Particle, kinetic and uid models for phototaxis, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), no. 1, 77-108. https://doi.org/10.3934/dcdsb.2009.12.77
  10. Z. He, On the existence of positive solutions of p-Laplacian difference equations, J. Comput. Appl. Math. 161 (2003), no. 1, 193-201. https://doi.org/10.1016/j.cam.2003.08.004
  11. D. Q. Jiang, D. O'Regan, and R. P. Agarwal, A generalized upper and lower solution method for singular discrete boundary value problems for the one-dimensional p-Laplacian, J. Appl. Anal. 11 (2005), no. 1, 35-47.
  12. J.-H. Kim, The (p, $\omega$)-Laplacian operators on nonlinear networks, Ph.D. Thesis, University of Sogang at Korea.
  13. J.-H. Kim, J.-H. Park, and J. Y. Lee, Multiple positive solutions for discrete p-Laplacian equations with potential term, Appl. Anal. Discrete Math. 7 (2013), no. 2, 327-342. https://doi.org/10.2298/AADM130612012K
  14. J.-H. Park and S.-Y. Chung, The Dirichlet boundary value problems for p-Schrodinger operators on finite networks, J. Difference Equ. Appl. 17 (2011), no. 5, 795-811. https://doi.org/10.1080/10236190903376204
  15. J.-H. Park and S.-Y. Chung, Positive solutions for discrete boundary value problems involving the p-Laplacian with potential terms, Comput. Math. Appl. 61 (2011), no. 1, 17-29. https://doi.org/10.1016/j.camwa.2010.10.026
  16. J.-H. Park, J.-H. Kim, and S.-Y. Chung, The p-Schrodinger equations on finite networks, Publ. Res. Inst. Math. Sci. 45 (2009), no. 2, 363-381. https://doi.org/10.2977/prims/1241553123
  17. V. Ta, S. Bougleux, A. Elmoataz, and O. Lezoray, Nonlocal anisotropic discrete regularization for image, data filtering and clustering, Tech. Rep., Univ. Caen, Caen, France, 2007.
  18. N. Trinajstic, Chemical Graph Theory, Second ed., CRC Press, Boca Raton, FL, 1992.
  19. N. Trinajstic, D. Babic, S. Nikolic, D. Plavsic, D. Amic, and Z. Mihalic, The Laplacian matrix in chemistry, J. Chem. Inf. Comput. Sci. 34 (1994), 368-376. https://doi.org/10.1021/ci00018a023

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