References
- Kirchhoff G. Mechanik. Leipzig: Teubner; 1983.
- Lions JL. On some questions in boundary value problems of mathematical physics. North-Holland Math Stud. 1978;30:284–346. doi: https://doi.org/10.1016/S0304-0208(08)70870-3
- Chipot M, Lovat B. Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 1997;30(7):4619–4627. doi: https://doi.org/10.1016/S0362-546X(97)00169-7
- Arosio A, Pannizi S. On the well-posedness of the Kirchhoff string. Trans Amer Math Soc. 1996;348:305–330. doi: https://doi.org/10.1090/S0002-9947-96-01532-2
- Cavalcanti MM, Cavacanti VN, Soriano JA. Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation. Adv Differ Equ. 2001;6:701–730.
- Corrêa FJSA, Figueiredo GM. On a elliptic equation of p-Kirchhoff type via variational methods. Bull Aust Math Soc. 2006;74(2):263–277. doi: https://doi.org/10.1017/S000497270003570X
- A. Corrêa FJS, Nascimento RG. On a nonlocal elliptic system of p-Kirchhoff type under Neumann boundary condition. Math Comput Model. 2008;49(3-4):598–604. doi:https://doi.org/10.1016/j.mcm.2008.03.013.
- Dia G. Existence of solutions for nonlocal elliptic systems with nonstandard growth conditions. Electron J Differ Equ. 2011;2011(137):1–13.
- Heidarkhani S, Henderson J. Infinitely many solutions for nonlocal elliptic systems of (p1,…,pn)-Kirchhoff type. Electron J Differ Equ. 2012;2012(69):1–15.
- Miao Q. Infinitely many solutions for nonlocal elliptic systems of (p1(x),…,pn(x))-Kirchhoff type. Math Meth Appl Sci. 2016;39:2325–2333. doi: https://doi.org/10.1002/mma.3642
- Perera K, Zhang Z. Nontrivial solutions of Kirchhof- type problems via the Yang index. J Differ Equ. 2006;221:246–255. doi: https://doi.org/10.1016/j.jde.2005.03.006
- Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math. 1983;36:437–477. doi: https://doi.org/10.1002/cpa.3160360405
- Bonder JF, Silva A. Concentration-compactness principal for variable exponent space and applications. Electron J Differ Equ. 2010;141:1–18.
- Fu YQ. The principle of concentration compactness in Lp(x) spaces and its application. Nonlinear Anal. 2009;71:1876–1892. doi: https://doi.org/10.1016/j.na.2009.01.023
- Benbernou S, Gala S, Ragusa MA. On the regularity criteria for the 3D magnetohydrodynamic equations via two components in terms of BMO space. Math Methods Appl Sci. 2014;37(15):2320–2325. doi: https://doi.org/10.1002/mma.2981
- Chen YM, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math. 2006;66:1383–1406. doi: https://doi.org/10.1137/050624522
- Diening L, Harjulehto P, Hästö P, et al. Lebesgue and Sobolev spaces with variable exponents. Heidelberg: Springer-Verlag; 2011. (Lecture Notes in Mathematics; 2017).
- Gala S, Liu Q, Ragusa MA. A new regularity criterion for the nematic liquid crystal flows. Appl Anal. 2012;91(9):1741–1747. doi: https://doi.org/10.1080/00036811.2011.581233
- Gala S, Ragusa MA. Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces. Appl Anal. 2016;95(6):1271–1279. doi: https://doi.org/10.1080/00036811.2015.1061122
- Halsey TC. Electrorheological fluids. Science. 1992;258:761–766. doi: https://doi.org/10.1126/science.258.5083.761
- Ragusa MA, Tachïkawa A. Regularity for minimizers for functionals of double phase with variable exponents. Adv Nonlinear Anal. 2020;9:710–728. doi: https://doi.org/10.1515/anona-2020-0022
- Ružika M. Flow of shear dependent electro-rheological fluids. C R Acad Sci Paris Ser I. 1999;329:393–398. doi: https://doi.org/10.1016/S0764-4442(00)88612-7
- Ružika M. Electro-rheological fluids: modeling and mathematical theory. Berlin: Springer; 2000. (Lecture notes in mathematics).
- Alves CO, Barreiro JP. Existence and multiplicity of solutions for a p(x)-Laplacian equation with critical growth. J Math Anal Appl. 2013;403:143–154. doi: https://doi.org/10.1016/j.jmaa.2013.02.025
- Hurtado EJ, Miyagaki OH, Rodrigues RS. Existence and asymptotic behaviour for a Kirchhof- type equation with variable critical growth exponent. Milan J Math. 2010;77:127–150.
- Lalilia H, Tasa S, Djellitb A. Existence of solutions for critical systems with variable exponents. Math Model Anal. 2018;23(4):596–610. doi: https://doi.org/10.3846/mma.2018.036
- Tsouli N, Haddaoui M, Hssini EM. Multiple solutions for a critical p(x)-Kirchhof- type equations. Bol Soc Paran Mat. 2020;38(4):197–211. doi: https://doi.org/10.5269/bspm.v38i4.37697
- Zhang X, Fu Y. Solutions of p(x)-Laplacian equations with critical exponent and perturbations in RN. Electron J Differ Equ. 2012;2012(120):1–14.
- Kováčik O, Rákosník J. On spaces Lp(x)(Ω) and W1,p(Ω). Czechoslovak Math J. 1991;41:592–618.
- Fan X, Zhao D. On the spaces Lp(x)(Ω) and Wm,p(Ω). J Math Anal Appl. 2001;263:424–446. doi: https://doi.org/10.1006/jmaa.2000.7617
- Edmunds DE, Rakosnik J. Sobolev embeddings with variable exponent. Studia Math. 2000;143:267–293. doi: https://doi.org/10.4064/sm-143-3-267-293
- Chems Eddine N, Idrissi AA. Multiple solutions to a (p1(x),…,pn(x))-Laplacian-type systems in unbounded domain. Azerbaijan J Math. 2020;10(1):3–20.
- DiBenedetto E. Degenerate parabolic equations. New York (NY): Springer-Verlag; 1993.
- Bonder JF, Martínez S, Rossi JD. Existence results for gradient elliptic systems with nonlinear boundary conditions. NoDEA. 2007;14(1-2):153–179. doi: https://doi.org/10.1007/s00030-007-5015-2
- Jebri Y. The mountain pass theorem: variants, generalizations, and some applications. New York (NY): Cambridge University Press; 2003.