Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problems
Abstract
This paper is concerned with the existence and multiplicity of solutions for a class of $p(x)$-Kirchhoff type equations with Neumann boundary condition. Our technical approach is based on variational methods.
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M. Allaoui, A. R, El Amrouss, A. Ourraoui, Existence and multiplicity of solutions for a Steklove problem involving the p(x)-Laplacian operator, Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 132, pp. 1-12.
M. Allaoui, A. R, El Amrouss, A. Ourraoui, Existence and uniqueness of solution for p(x)- Laplacian problems, Bol. Soc. Paran. Mat, V. 33 1 (2015), 225-232.
C. O. Alves, F. J. S. A. Corrêa, T. F. MA, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85-93.
G. A. Afrouzi, A. Hadjian, S. Heidarkhani, Steklove problems involving the p(x)-Laplacian, Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 134, pp. 1-11.
S. Antontsev, S. Shmarev, Handbook of Differential Equations, Stationary Partial Differ. Equ. 3 (2006). Chap. 1.
A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Am. Math. Soc. 348 (1996) 305-330.
M. Bocea, M. Mihailescu, Γ-convergence of power-law functionals with variable exponents, Nonlinear Anal. 73 (2010) 110-121.
M. Bocea, M. Mihailescu, C. Popovici, On the asymptotic behavior of variable exponent power-law functionals and applications, Ric. Mat. 59 (2010) 207-238.
G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012) 2992-3007.
G. Bonanno, A. Sciammetta, Existence and multiplicity results to Neumann problems for elliptic equations involving the p-Laplacian, J. Math. Anal. Appl. 390 (2012) 59-67.
F. J. S. A. Corrêa, G. M. Figueiredo, On a elliptic equation of p-Kirchhoff type via variational methods, Bull. Aust. Math. Soc. 74 (2006) 263-277.
M. M. Cavalcanti, V. N. Cavalcanti, J. A. Soriano, Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001) 701-730
Y. Chen, S. Levine, R. Rao, Variable exponent, linear growth functionals in image restoration, SIAMJ. Appl. Math. 66 (2006) 1383-1406.
F. Colasuonno, P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011) 5962-5974.
F. Cammaroto, L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator, Nonlinear Anal. 74 (2011) 1841-1852.
P. D'Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic date. Invent. Math. 108 (1992) 447-462.
G. Dai, Infinitely many solutions for a p(x)-Laplacian equation in RN , Nonlinear Anal. 71 (2009) 1133-1139.
G. Dai, Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)-Laplacian, App. Anal. (2011) 1-20.
G. Dai, D. Liu, Infinitely many positive solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009) 704-710.
G. Dai, R. Ma, Solutions for a p(x)-Kirchhoff type equation with Neumann boundary data, Nonlinear Anal. RWA 12 (2011) 2666-2680.
G. Dai, R. Ma, Solutions for a p(x)-Kirchhoff type equation with Neumann boundary data, Nonlinear Analysis: Real World Applications 12 (2011) 2666-2680.
G. Dai, J. Wei, Infinitely many non-negative solutions for a p(x)-Kirchhoff-type problem with Dirichlet boundary condition, Nonlinear Anal. 73 (2010) 3420-3430.
G. D'Aguì, A. Sciammetta; Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions, Nonlinear Anal., 75 (2012), 5612-5619.
X. Fan, X. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in RN, Nonlinear Anal. 59 (2004) 173-188.
M. Ferrara, S. Heidarkhani, Multiple solutions for perturbed p-Laplacian boundary-value problems with impulsive effects, Electronic journal of differential equations, n. 2014, 2014, pp. 1-14, ISSN: 1072-6691.
M. Ferrara, S. Khademloob, S. Heidarkhani, Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems, Applied mathematics and computation, n. 234, 2014, pp. 316-325, ISSN: 1873-5649.
X. Fan, D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω) J. Math. Anal. Appl. 263 (2001) 424-446.
X. Fan, Q. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843-1852.
Y. Fu, X. Zhang, A multiplicity result for p(x)-Laplacian problem in RN , Nonlinear Anal. 70 (2009) 2261-2269.
El. M. Hssini, M. Massar, M. Talbi and N. Tsouli, Infinitely many solutions for nonlocal elliptic p-Kirchhoff type equation under Neumann boundary condition, Int. Journal of Math. Analysis, Vol. 7, 2013, no. 21, 1011-1022.
J. L. Lions, On some questions in boundary value problems of mathematical physical .In : de la Penha, GM, Medeiros, LAJ (eds.) Proceedings of International Symposium on Continuum Mechanics and Partial Differential Equations, Rio deJanerro 1977. Math. Stud., vol. 30 (1978), pp. 284-346. North-Holland, Amsterdam.
R. Ma, G. Dai and C. Gao, Existence and multiplicity of positive solutions for a class of p(x)-Kirchhoff type equations, Boundary Value Problems 2012, 2012:16, 1-16.
M. Massar, Existence and multiplicity results for nonlocal elliptic problem, Electronic Journal of Differential Equations, Vol. 2013 (2013), No., pp. 1-14.
M. Massar, M. Talbi, N. Tsouli, Multiple solutions for nonlocal system of (p(x), q(x))- Kirchhoff type, Appl. Math. Comput. 242 (2014) 216-226.
M. Mihailescu, On a class of nonlinear problems involving a p(x)-Laplace type operator, Czechoslovak Math. J. 58 (2008) 155-172.
G. Kirchhoff, Mechanik, Teubner, leipzig, Germany, 1883.
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. Math. Soc., Providence, RI, 1986.
M. Růžička, Flow of shear dependent electrorheological fluids, CR Math. Acad. Sci. Paris 329 (1999) 393-398.
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002.
E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II/B, Berlin, Heidelberg, New York, 1985.
V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987) 33-66.
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