Browse > Article
http://dx.doi.org/10.4134/CKMS.c200132

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)
Publication Information
Communications of the Korean Mathematical Society / v.36, no.1, 2021 , pp. 51-62 More about this Journal
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
(p(x), q(x))-Laplacian systems; variational methods; critical points;
Citations & Related Records
연도 인용수 순위
  • Reference
1 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   DOI
2 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.
3 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   DOI
4 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   DOI
5 M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. (2020), 10 pages.
6 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.
7 Q. Miao, Multiple solutions for nonlocal elliptic systems involving p(x)-biharmonic operator, Mathematics 7 (2019), no. 8, 756.   DOI
8 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   DOI
9 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.
10 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
11 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   DOI
12 M. A. Ragusa, Elliptic boundary value problem in vanishing mean oscillation hypothesis, Comment. Math. Univ. Carolin. 40 (1999), no. 4, 651-663.
13 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   DOI
14 M. A. Ragusa and A. Razani, Weak solutions for a system of quasilinear elliptic equations, Contrib. Math. 1 (2020) 11-26.
15 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   DOI
16 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   DOI
17 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   DOI
18 A. Razani, Shock waves in gas dynamics, Surv. Math. Appl. 2 (2007), 59-89.
19 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   DOI
20 A. Razani, Chapman-Jouguet travelling wave for a two-steps reaction scheme, Ital. J. Pure Appl. Math. 39 (2018), 544-553.
21 A. Razani, Subsonic detonation waves in porous media, Phys. Scr. 94 (2019), no. 085209, 6 pages.
22 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   DOI
23 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   DOI
24 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   DOI
25 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   DOI
26 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   DOI
27 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.
28 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   DOI
29 C. Cowan and A. Razani, Singular solutions of a Lane-Emden system, Discrete and Continuous Dynamical Systems accepted (2020), doi: 10.3934/dcds.2020291.   DOI
30 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
31 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   DOI
32 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.
33 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   DOI
34 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.
35 O. Kovacik and J. Rakosnik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41(116) (1991), no. 4, 592-61   DOI
36 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   DOI