On a class of fractional p(x) -Kirchhoff type problems

References

• Applebaum D. Lévy processes-from probability to fnance quantum groups. Not Am Math Soc. 2004;51: 1336–1347.
   Google Scholar
• Caffarelli L. Nonlocal equations, drifts and games. Nonlinear partial differential equations. New York: Springer; 2012. p. 37–52. (Abel symposia; vol. 7).
   Google Scholar
• Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A. 2000;268:298–305. doi: 10.1016/S0375-9601(00)00201-2
   Web of Science ®Google Scholar
• Servadei R, Valdinoci E. Mountain pass solutions for non-local elliptic operators. J Math Anal Appl. 2012;389(2):887–898. doi: 10.1016/j.jmaa.2011.12.032
   Web of Science ®Google Scholar
• Servadei R, Valdinoci E. Variational methods for non-local operators of elliptic type. Discrete Contin Dyn Syst. 2013;33(5):2105–2137.
   Web of Science ®Google Scholar
• Servadei R, Valdinoci E. Weak and viscosity solutions of the fractional Laplace equation. Publ Mat. 2014;58:133–154. doi: 10.5565/PUBLMAT_58114_06
   Web of Science ®Google Scholar
• Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136(5):521–573. doi: 10.1016/j.bulsci.2011.12.004
   Web of Science ®Google Scholar
• Zhikov VV. Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv. 1987;29:33–66. doi: 10.1070/IM1987v029n01ABEH000958
   Google Scholar
• Acerbi E, Mingione G. Gradient estimate for the p(x)-Laplacian system. J Reine Angew Math. 2005;2005(584): 117–148. doi: 10.1515/crll.2005.2005.584.117
   Google Scholar
• Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image processing. SIAM J Appl Math. 2006;66:1383–1406. doi: 10.1137/050624522
   Web of Science ®Google Scholar
• Chen W. Multiplicity of solutions for a fractional Kirchhoff type problem. Commun Pure Appl Anal. 2015;14:2009–2020. doi: 10.3934/cpaa.2015.14.2009
   Web of Science ®Google Scholar
• Ferrara M, Xiang M, Zhang B. Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities. Proc R Soc A. 2015;471:20150034.
   Google Scholar
• Fiscella A, Valdinoci E. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 2014;94:156–170. doi: 10.1016/j.na.2013.08.011
   Web of Science ®Google Scholar
• Cammaroto F, Vilasi L. Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator. Nonlinear Anal. 2011;74:1841–1852. doi: 10.1016/j.na.2010.10.057
   Web of Science ®Google Scholar
• Xiang M, Zhang B. Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition. Complex Var Elliptic Equ. 2015;60:1277–1287. doi: 10.1080/17476933.2015.1005612
   Web of Science ®Google Scholar
• Kirchhoff G. Mechanik. Leipzig: Teubner; 1883.
   Google Scholar
• Lions JL. On some questions in boundary value problems of mathematical physics. In: de la Penha GM, Medeiros LA, editors. Proceedings of international symposium on continuum mechanics and partial differential equations, Rio de Janeiro, 1977. NorthHolland; 1978. p. 284–346. (Math. stud.; 30).
   Google Scholar
• Arosio A, Panizzi S. On the well-posedness of the Kirchhoff string. Trans Amer Math Soc. 1996;348(1):305–330. doi: 10.1090/S0002-9947-96-01532-2
   Web of Science ®Google Scholar
• Cavalcanti MM, Domingos Cavalcanti VN, Soriano JA. Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation. Adv Differ Equ. 2001;6:701–730.
   Google Scholar
• Corrêa FJ, Menezes SD, Ferreira J. On a class of problems involving a nonlocal operator. Appl Math Comput. 2004;147:475–489.
   Web of Science ®Google Scholar
• Corrêa FJSA, Figueiredo GM. On an elliptic equation of p-Kirchhoff type via variational methods. Bull Aust Math Soc. 2006;74:263–277. doi: 10.1017/S000497270003570X
   Web of Science ®Google Scholar
• Corrêa FJSA, Figueiredo GM. On a p-Kirchhoff equation via Krasnoselskii's genus. Appl Math Lett. 2009;22:819–822. doi: 10.1016/j.aml.2008.06.042
   Web of Science ®Google Scholar
• Dreher M. The Kirchhoff equation for the p-Laplacian. Rend Semin Mat Univ Politec Torino. 2006;64:217–238.
   Google Scholar
• Dai G, Hao R. Existence of solutions for a p(x)-Kirchhoff-type equation. J Math Anal Appl. 2009;359:275–284. doi: 10.1016/j.jmaa.2009.05.031
   Web of Science ®Google Scholar
• Chung NT. Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities. Electron J Qual Theory Differ Equ. 2012;2012(42):1–13. doi: 10.14232/ejqtde.2012.1.42
   Google Scholar
• Colasuonno F, Pucci P. Multiplicity of solutions for p(x)-polyharmonic Kirchhoff equations. Nonlinear Anal. 2011;74:5962–5974. doi: 10.1016/j.na.2011.05.073
   Web of Science ®Google Scholar
• Fan XL. On nonlocal p(x)-Laplacian Dirichlet problems. Nonlinear Anal. 2010;72:3314–3323. doi: 10.1016/j.na.2009.12.012
   Web of Science ®Google Scholar
• Afrouzi GA, Mirzapour M. Eigenvalue problems for p(x)-Kirchhoff type equations. Electron J Differ Equ. 2013;2013(253):1–10.
   Google Scholar
• Afrouzi GA, Mirzapour M, Chung NT. Existence and multiplicity of solutions for a p(x)-Kirchhoff type equation. Rend Semin Mat Univ Padova. 2016;136:95–109. doi: 10.4171/RSMUP/136-8
   Web of Science ®Google Scholar
• Fan XL, Zhao D. On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). J Math Anal Appl. 2001;263:424–446. doi: 10.1006/jmaa.2000.7617
   Web of Science ®Google Scholar
• Kováčik O, Rákosník J. On spaces Lp(x)(Ω) and Wm,p(x)(Ω). Czechoslovak Math J. 1991;41(4):592–618.
   Web of Science ®Google Scholar
• Azroul E, Benkirane A, Shimi M. Eigenvalue problems involving the fractional p(x)-Laplacian operator. Adv Oper Theory. 2019;4(2):539–555. DOI:10.15352/aot.1809-1420 doi: 10.15352/aot.1809-1420
   Web of Science ®Google Scholar
• Bahrouni A, Rădulescu V. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete Contin Dyn Syst. 2018;11:379–389. doi: 10.3934/dcdss.2018021
   Web of Science ®Google Scholar
• Kaufmann U, Rossi JD, Vidal R. Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians. Electron J Qual Theory Differ Equ. 2017;76:1–10. doi: 10.14232/ejqtde.2017.1.76
   Web of Science ®Google Scholar
• Servadei R, Valdinoci E. Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators. Rev Mat Iberoam. 2013;29(3):1091–1126. doi: 10.4171/RMI/750
   Web of Science ®Google Scholar
• Chen W, Deng S. The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities. Nonlinear Anal. 2016;27:80–92. doi: 10.1016/j.nonrwa.2015.07.009
   Web of Science ®Google Scholar
• Ambrosetti A, Rabinowitz PH. Dual variational methods in critical points theory and applications. J Funct Anal. 1973;14:349–381. doi: 10.1016/0022-1236(73)90051-7
   Google Scholar
• Struwe M. Variational methods: applications to nonlinear partial differential equatios and Hamiltonian systems. Berlin: Springer-Verlag; 1996.
   Google Scholar
• Ekeland I. On the variational principle. J Math Anal Appl. 1974;47:324–353. doi: 10.1016/0022-247X(74)90025-0
   Web of Science ®Google Scholar