Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

INFINITELY MANY SOLUTIONS FOR (p(x), q(x))-LAPLACIAN-LIKE SYSTEMS

  • Heidari, Samira (Department of Pure Mathematics Faculty of Science Imam Khomeini International University) ;
  • Razani, Abdolrahman (Department of Pure Mathematics Faculty of Science Imam Khomeini International University)
  • Received : 2020.04.19
  • Accepted : 2020.08.31
  • Published : 2021.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Variational method has played an important role in solving problems of uniqueness and existence of the nonlinear works as well as analysis. It will also be extremely useful for researchers in all branches of natural sciences and engineers working with non-linear equations economy, optimization, game theory and medicine. Recently, the existence of infinitely many weak solutions for some non-local problems of Kirchhoff type with Dirichlet boundary condition are studied [14]. Here, a suitable method is presented to treat the elliptic partial derivative equations, especially (p(x), q(x))-Laplacian-like systems. This kind of equations are used in the study of fluid flow, diffusive transport akin to diffusion, rheology, probability, electrical networks, etc. Here, the existence of infinitely many weak solutions for some boundary value problems involving the (p(x), q(x))-Laplacian-like operators is proved. The method is based on variational methods and critical point theory.

Keywords

References

  1. G. A. Afrouzi and S. Shokooh, Existence of infinitely many solutions for quasilinear problems with a p(x)-biharmonic operator, Electron. J. Differential Equations 2015 (2015), No. 317, 14 pp.
  2. G. Bonanno and G. M. Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009 (2009), Art. ID 670675, 20 pp. https://doi.org/10.1155/2009/670675
  3. C. Cowan and A. Razani, Singular solutions of a p-Laplace equation involving the gradient, J. Differential Equations 269 (2020), no. 4, 3914-3942. https://doi.org/10.1016/j.jde.2020.03.017
  4. C. Cowan and A. Razani, Singular solutions of a Lane-Emden system, Discrete and Continuous Dynamical Systems accepted (2020), doi: 10.3934/dcds.2020291.
  5. L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8
  6. X. Fan, Q. Zhang, and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), no. 2, 306-317. https://doi.org/10.1016/j.jmaa.2003.11.020
  7. X. Fan and D. Zhao, On the generalized Orlicz-Sobolev spaces Wk,p(x)(Ω), J. Gansu Educ. College 12 (1998), no. 1, 1-6.
  8. X. Fan and D. Zhao. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), no. 2, 424-446. https://doi.org/10.1006/jmaa.2000.7617
  9. A. Ghelichi and M. Alimohammady, Existence of bound states for non-local fourth-order Kirchhoff systems, Comput. Methods Differ. Equ. 7 (2019), no. 3, 418-433.
  10. S. Heidarkhani, M. Ferrara, A. Salari, and G. Caristi, Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equ. 61 (2016), no. 11, 1494-1516. https://doi.org/10.1080/17476933.2016.1182520
  11. O. Kovacik and J. Rakosnik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41(116) (1991), no. 4, 592-61 https://doi.org/10.21136/CMJ.1991.102493
  12. M. Makvand Chaharlang, M. A. Ragusa, and A. Razani, A sequence of radially symmetric weak solutions for some nonlocal elliptic problem in ℝN , Mediterr. J. Math. 17 (2020), no. 2, Art. 53, 12 pp. https://doi.org/10.1007/s00009-020-1492-x
  13. M. Makvand Chaharlang and A. Razani, A fourth order singular elliptic problem involving p-biharmonic operator, Taiwanese J. Math. 23 (2019), no. 3, 589-599. https://doi.org/10.11650/tjm/180906
  14. M. Makvand Chaharlang and A. Razani, Existence of infinitely many solutions for a class of nonlocal problems with Dirichlet boundary condition, Commun. Korean Math. Soc. 34 (2019), no. 1, 155-167. https://doi.org/10.4134/CKMS.c170456
  15. M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. (2020), 10 pages.
  16. M. Massar, E. M. Hssini, and N. Tsouli, Infinitely many solutions for class of Navier boundary (p, q)-biharmonic systems, Electron. J. Differential Equations 2012 (2012), No. 163, 9 pp.
  17. Q. Miao, Multiple solutions for nonlocal elliptic systems involving p(x)-biharmonic operator, Mathematics 7 (2019), no. 8, 756. https://doi.org/10.3390/math7080756
  18. M. R. Mokhtarzadeh, M. R. Pournaki, and A. Razani, A note on periodic solutions of Riccati equations, Nonlinear Dynam. 62 (2010), no. 1-2, 119-125. https://doi.org/10.1007/s11071-010-9703-9
  19. M. R. Mokhtarzadeh, M. R. Pournaki, and A. Razani, An existence-uniqueness theorem for a class of boundary value problems, Fixed Point Theory 13 (2012), no. 2, 583-591.
  20. V. D. Radulescu and D. D. Repovs, Partial differential equations with variable exponents, Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2015. https://doi.org/10.1201/b18601
  21. M. A. Ragusa, Elliptic boundary value problem in vanishing mean oscillation hypothesis, Comment. Math. Univ. Carolin. 40 (1999), no. 4, 651-663.
  22. M. A. Ragusa, Necessary and sufficient condition for a V MO function, Appl. Math. Comput. 218 (2012), no. 24, 11952-11958. https://doi.org/10.1016/j.amc.2012.06.005
  23. M. A. Ragusa and A. Razani, Weak solutions for a system of quasilinear elliptic equations, Contrib. Math. 1 (2020) 11-26.
  24. A. Razani, Weak and strong detonation profiles for a qualitative model, J. Math. Anal. Appl. 276 (2002), no. 2, 868-881. https://doi.org/10.1016/S0022-247X(02)00459-6
  25. A. Razani, Chapman-Jouguet detonation profile for a qualitative model, Bull. Austral. Math. Soc. 66 (2002), no. 3, 393-403. https://doi.org/10.1017/S0004972700040259
  26. A. Razani, Existence of Chapman-Jouguet detonation for a viscous combustion model, J. Math. Anal. Appl. 293 (2004), no. 2, 551-563. https://doi.org/10.1016/j.jmaa.2004.01.018
  27. A. Razani, On the existence of premixed laminar flames, Bull. Austral. Math. Soc. 69 (2004), no. 3, 415-427. https://doi.org/10.1017/S0004972700036194
  28. A. Razani, Shock waves in gas dynamics, Surv. Math. Appl. 2 (2007), 59-89.
  29. A. Razani, An existence theorem for an ordinary differential equation in Menger probabilistic metric space, Miskolc Math. Notes 15 (2014), no. 2, 711-716. https://doi.org/10.18514/mmn.2014.640
  30. A. Razani, Chapman-Jouguet travelling wave for a two-steps reaction scheme, Ital. J. Pure Appl. Math. 39 (2018), 544-553.
  31. A. Razani, Subsonic detonation waves in porous media, Phys. Scr. 94 (2019), no. 085209, 6 pages.
  32. B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), no. 1-2, 401-410. https://doi.org/10.1016/S0377-0427(99)00269-1
  33. M. M. Rodrigues, Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators, Mediterr. J. Math. 9 (2012), no. 1, 211-223. https://doi.org/10.1007/s00009-011-0115-y
  34. F. Safari and A. Razani, Existence of positive radial solution for Neumann problem on the Heisenberg group, Bound. Value Probl. 2020 (2020), Paper No. 88, 14 pp. https://doi.org/10.1186/s13661-020-01386-5
  35. F. Safari and A. Razani, Nonlinear nonhomogeneous Neumann problem on the Heisenberg group, Appl. Anal. 2020 (2020). https://doi.org/10.1080/00036811.2020.1807013
  36. S. Shokooh and A. Neirameh, Existence results of infinitely many weak solutions for p(x)-Laplacian-like operators, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 78 (2016), no. 4, 95-104.