On a Robin type problem involving 𝑝(𝑥)-Laplacian operator

References

[1] M. Allaoui, A. R. El Amrouss and A. Ourraoui, Existence of infinitely many solutions for a Steklov problem involving the p ⁢ ( x ) -Laplace operator, Electron. J. Qual. Theory Differ. Equ. 2014 (2014), Paper No. 20. 10.14232/ejqtde.2014.1.20Search in Google Scholar

[2] M. Allaoui and A. Ourraoui, Existence results for a class of p ⁢ ( x ) -Kirchhoff problem with a singular weight, Mediterr. J. Math. 13 (2016), no. 2, 677–686. 10.1007/s00009-015-0518-2Search in Google Scholar

[3] C. O. Alves and M. A. S. Souto, Existence of solutions for a class of problems in R N involving the p ⁢ ( x ) -Laplacian, Contributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl. 66, Birkhäuser, Basel (2006), 17–32. 10.1007/3-7643-7401-2_2Search in Google Scholar

[4] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. 10.1016/0022-1236(73)90051-7Search in Google Scholar

[5] G. Autuori, P. Pucci and M. C. Salvatori, Asymptotic stability for anisotropic Kirchhoff systems, J. Math. Anal. Appl. 352 (2009), no. 1, 149–165. 10.1016/j.jmaa.2008.04.066Search in Google Scholar

[6] M. Avci, Existence and multiplicity of solutions for Dirichlet problems involving the p ⁢ ( x ) -Laplace operator, Electron. J. Differential Equations 2013 (2013), Paper No. 14. Search in Google Scholar

[7] M. Avci and K. Suslu, On a Robin Problem in Orlicz–Sobolev Spaces, TWMS J. App. Eng. Math. 9 (2019), no. 2, 246–256. Search in Google Scholar

[8] A. Ayoujil, On the superlinear Steklov problem involving the p ⁢ ( x ) -Laplacian, Electron. J. Qual. Theory Differ. Equ. 2014 (2014), Paper No. 38. 10.14232/ejqtde.2014.1.38Search in Google Scholar

[9] P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. 7 (1983), no. 9, 981–1012. 10.1016/0362-546X(83)90115-3Search in Google Scholar

[10] M. Bocea and M. Mihăilescu, Γ-convergence of power-law functionals with variable exponents, Nonlinear Anal. 73 (2010), no. 1, 110–121. 10.1016/j.na.2010.03.004Search in Google Scholar

[11] M. Bocea, M. Mihăilescu and C. Popovici, On the asymptotic behavior of variable exponent power-law functionals and applications, Ric. Mat. 59 (2010), no. 2, 207–238. 10.1007/s11587-010-0081-xSearch in Google Scholar

[12] G. Bonanno and A. Chinnì, Discontinuous elliptic problems involving the p ⁢ ( x ) -Laplacian, Math. Nachr. 284 (2011), no. 5–6, 639–652. 10.1002/mana.200810232Search in Google Scholar

[13] B. Cekic and R. A. Mashiyev, Existence and localization results for p ⁢ ( x ) -Laplacian via topological methods, Fixed Point Theory Appl. 2010 (2010), Article ID 120646. 10.1155/2010/120646Search in Google Scholar

[14] G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), no. 2, 332–336. Search in Google Scholar

[15] J. Chabrowski and Y. Fu, Existence of solutions for p ⁢ ( x ) -Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), no. 2, 604–618. 10.1016/j.jmaa.2004.10.028Search in Google Scholar

[16] K. C. Chang, Critical Point Theory and Its Applications (in Chinese), Mod. Math. Ser., Shanghai Kexue Jishu Chubanshe, Shanghai, 1986. Search in Google Scholar

[17] N. T. Chung, On a class of p(x)-kirhhoff type problems with robin boundary conditions and indefinite weights, TWMS J. App. Eng. Math. 10 (2020), no. 2, 400–410. Search in Google Scholar

[18] F. Colasuonno and P. Pucci, Multiplicity of solutions for p ⁢ ( x ) -polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011), no. 17, 5962–5974. 10.1016/j.na.2011.05.073Search in Google Scholar

[19] D. G. Costa and C. A. Magalhães, Existence results for perturbations of the 𝑝-Laplacian, Nonlinear Anal. 24 (1995), no. 3, 409–418. 10.1016/0362-546X(94)E0046-JSearch in Google Scholar

[20] G. D’Aguì and A. Sciammetta, Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions, Nonlinear Anal. 75 (2012), no. 14, 5612–5619. 10.1016/j.na.2012.05.009Search in Google Scholar

[21] S.-G. Deng, Positive solutions for Robin problem involving the p ⁢ ( x ) -Laplacian, J. Math. Anal. Appl. 360 (2009), no. 2, 548–560. 10.1016/j.jmaa.2009.06.032Search in Google Scholar

[22] S.-G. Deng, Q. Wang and S. Cheng, On the p ⁢ ( x ) -Laplacian Robin eigenvalue problem, Appl. Math. Comput. 217 (2011), no. 12, 5643–5649. 10.1016/j.amc.2010.12.042Search in Google Scholar

[23] X. Fan and X. Han, Existence and multiplicity of solutions for p ⁢ ( x ) -Laplacian equations in R N , Nonlinear Anal. 59 (2004), no. 1–2, 173–188. 10.1016/S0362-546X(04)00254-8Search in Google Scholar

[24] X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces W k , p ⁢ ( x ) ⁢ ( Ω ) , J. Math. Anal. Appl. 262 (2001), no. 2, 749–760. 10.1006/jmaa.2001.7618Search in Google Scholar

[25] X.-L. Fan and Q.-H. Zhang, Existence of solutions for p ⁢ ( x ) -Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), no. 8, 1843–1852. 10.1016/S0362-546X(02)00150-5Search in Google Scholar

[26] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman–Lazer-type problem set on R N , Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787–809. 10.1017/S0308210500013147Search in Google Scholar

[27] O. Kováčik and J. Rákosník, On spaces L p ⁢ ( x ) and W k , p ⁢ ( x ) , Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. 10.21136/CMJ.1991.102493Search in Google Scholar

[28] S. B. Liu and S. J. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser.) 46 (2003), no. 4, 625–630. Search in Google Scholar

[29] A. Ourraoui, Multiplicity results for Steklov problem with variable exponent, Appl. Math. Comput. 277 (2016), 34–43. 10.1016/j.amc.2015.12.043Search in Google Scholar

[30] N. S. Papageorgiou and V. D. Rădulescu, Positive solutions of nonlinear Robin eigenvalue problems, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4913–4928. 10.1090/proc/13107Search in Google Scholar

[31] P. Pucci, Geometric description of the mountain pass critical points, Contemp. Math. 2 (2014), 469–471. 10.1007/978-3-0348-0847-7_9Search in Google Scholar

[32] A. Qian and C. Li, Infinitely many solutions for a Robin boundary value problem, Int. J. Differ. Equ. 2010 (2010), Article ID 548702. 10.1155/2010/548702Search in Google Scholar

[33] K. R. Rajagopal and M. Ruzicka, Mathematical modeling of electrorheological materials, Contin. Mech. Thermodyn. 13 (2001), no. 1, 59–78. 10.1007/s001610100034Search in Google Scholar

[34] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer, Berlin, 2000. 10.1007/BFb0104029Search in Google Scholar

[35] N. Tsouli and O. Darhouche, Existence and multiplicity results for nonlinear problems involving the p ⁢ ( x ) -Laplace operator, Opuscula Math. 34 (2014), no. 3, 621–638. 10.7494/OpMath.2014.34.3.621Search in Google Scholar

[36] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996. 10.1007/978-1-4612-4146-1Search in Google Scholar

[37] Z. Yucedag, Existence of solutions for p ⁢ ( x ) Laplacian equations without Ambrosetti–Rabinowitz type condition, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 3, 1023–1033. 10.1007/s40840-014-0057-1Search in Google Scholar

[38] A. Zang, p ⁢ ( x ) -Laplacian equations satisfying Cerami condition, J. Math. Anal. Appl. 337 (2008), no. 1, 547–555. 10.1016/j.jmaa.2007.04.007Search in Google Scholar

[39] J. Zhang and X. Xue, Multiple solutions of 𝑝-Laplacian with Neumann and Robin boundary conditions for both resonance and oscillation problem, Bound. Value Probl. 2011 (2011), Article ID 214289. 10.1186/1687-2770-2011-214289Search in Google Scholar

[40] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710; translation in Math. USSR-Izv. 29 (1987), no. 1, 33–66. 10.1070/IM1987v029n01ABEH000958Search in Google Scholar

[41] W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001), no. 3, 343–358. 10.1007/s002290170032Search in Google Scholar