Existence of multiple solutions for a p(x)- biharmonic equation
Keywords:
Neumann problem; p(x)- biharmonic operator; critical points.
Abstract
The aim of this paper is to obtain at least three solutions for a Neumann problem involving the p(x)-biharmonic operator. The main tool used for obtaining our result is a three critical points theorem established by Ricceri.
Downloads
Download data is not yet available.
References
[1] X.L.Fan, D.Zhao, On the space ( ) ( ) p x L and ( ) ( ) mp x W , J.Math.Anal.Appl. 263(2001), 424-446.
[2] M.Mihailescu, Electrorheological fluids: modeling and mathematical theory, J.Springer-Verlag, Berlin. 2000.
[3] V.V.Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, J.Izv.Akad.Nauk SSSR Ser. Mat., 50(1986),675-710.
[4] G.Bonanno, B.Di Bella, A boundary value problem for fourth-order elastic beam equations, J.Math.Anal.Appl. 343(2) (2008),1116-1176.
[5] A.M.Micheletti, A.Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, J.Nonlinear Anal 31(7) (1998),895-908.
[6] M.Mihailescu, V.Radulescu, A multiplicity result for nonlinear degenerate problem arising in the theory of electrorheological fluids, J.Proc.R.Soc.Lond.Ser.A 462(2006), 2625-2641.
[7] J.Musielak,Orlicz Space and Modular Space, J.Lecture Notes in Mathematics.1034(1983).
[8] B.Ricceri, A three critical points theorem revisited, J.Nonlinear Anal. 70(2009), 3084-3089.
[9] B.Ricceri, Existence of three solutions for a class of elliptic eigenvalue problem, J.Math.Compact.Modelling. 32(2000), 1485-1494.
[10] Xiayang Shi, Xuanhao Ding, Existence and multiplicity of solutions for a general p(x)-Laplacian Neumann problem, J.Nonlinear Anal. 70(2009), 3715-3720.
[11] Abdel Rachid El Amrouse, Anass Ourraoui, Existence of solutions for a boundary problem involving p(x)-biharmonic operator, J.Bol. Soc. Paran. Mat. 31,1(2013), 179-192.
[12] F.Cammaroto, A.Chinni, B.Di Bella, Multiple solutions for a Neumann problem involving the p(x)-Laplacian, J.Nonlinear Anal. 71(2009), 4486-4492.
[13] LIN LI, LING FING, WEN-WU PAN, Existence of multiple solutions for a p(x)-biharmonic equation, J.Electronic Journal of Differential Equations, 2013, No.139(2013), 1-10.
[14] Lin-Lin Wang, Yong-Hong Fan, Wei-Gao Ge, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, J.Nonlinear Anal. 71(2009), 4259-4270.
[15] A.Zang, Y.Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, J.Nonlinear Anal. 69(2008), 3629-3636.
[16] Shapour.Heidarkhani, Yu Tian, Multiplicity results for a class of gradient systems depending on two parameters, J.Nonlinear Anal. 73(2010), 547-554.
[2] M.Mihailescu, Electrorheological fluids: modeling and mathematical theory, J.Springer-Verlag, Berlin. 2000.
[3] V.V.Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, J.Izv.Akad.Nauk SSSR Ser. Mat., 50(1986),675-710.
[4] G.Bonanno, B.Di Bella, A boundary value problem for fourth-order elastic beam equations, J.Math.Anal.Appl. 343(2) (2008),1116-1176.
[5] A.M.Micheletti, A.Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, J.Nonlinear Anal 31(7) (1998),895-908.
[6] M.Mihailescu, V.Radulescu, A multiplicity result for nonlinear degenerate problem arising in the theory of electrorheological fluids, J.Proc.R.Soc.Lond.Ser.A 462(2006), 2625-2641.
[7] J.Musielak,Orlicz Space and Modular Space, J.Lecture Notes in Mathematics.1034(1983).
[8] B.Ricceri, A three critical points theorem revisited, J.Nonlinear Anal. 70(2009), 3084-3089.
[9] B.Ricceri, Existence of three solutions for a class of elliptic eigenvalue problem, J.Math.Compact.Modelling. 32(2000), 1485-1494.
[10] Xiayang Shi, Xuanhao Ding, Existence and multiplicity of solutions for a general p(x)-Laplacian Neumann problem, J.Nonlinear Anal. 70(2009), 3715-3720.
[11] Abdel Rachid El Amrouse, Anass Ourraoui, Existence of solutions for a boundary problem involving p(x)-biharmonic operator, J.Bol. Soc. Paran. Mat. 31,1(2013), 179-192.
[12] F.Cammaroto, A.Chinni, B.Di Bella, Multiple solutions for a Neumann problem involving the p(x)-Laplacian, J.Nonlinear Anal. 71(2009), 4486-4492.
[13] LIN LI, LING FING, WEN-WU PAN, Existence of multiple solutions for a p(x)-biharmonic equation, J.Electronic Journal of Differential Equations, 2013, No.139(2013), 1-10.
[14] Lin-Lin Wang, Yong-Hong Fan, Wei-Gao Ge, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, J.Nonlinear Anal. 71(2009), 4259-4270.
[15] A.Zang, Y.Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, J.Nonlinear Anal. 69(2008), 3629-3636.
[16] Shapour.Heidarkhani, Yu Tian, Multiplicity results for a class of gradient systems depending on two parameters, J.Nonlinear Anal. 73(2010), 547-554.
Published
2015-12-17
How to Cite
Wang, X., & Tian, Y. (2015). Existence of multiple solutions for a p(x)- biharmonic equation. Journal of Progressive Research in Mathematics, 6(1), 722-733. Retrieved from https://scitecresearch.com/journals/index.php/jprm/article/view/522
Issue
Section
Articles
Copyright (c) 2015 Journal of Progressive Research in Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.