Existence and uniqueness of solutions for Steklov problem with variable exponent

In this article, we give some results on the existence and uniqueness of solutions concerned a class of elliptic problems involving $p(x)-$Laplacian with Steklov boundary condition. We give also some sufficient conditions to assure the existence of a positive solution.

Keywords:

Uniqueness of solution, $p(x)-$Laplacian operator Boundary value problem,

PDF

___

[1] E. Acerbi and G. Mingione, Regularity results for stationary electrorheological ?uids, Arch. Ration. Mech. Anal. 164 (2002) 213-259.
[2] M.V. Abdelkader, A. Ourraoui, Existence And Uniqueness Of Weak Solution For p-Laplacian Problem In R N , Applied Mathematics E-Notes, 13(2013), 228-233.
[3] G.A. Afrouzi, A. Hadjian, S. Heidarkhani, S. Shokooh, In?nitely many solutions for Steklov problems associated to non- homogeneous differential operators through Orlicz-Sobolev spaces. Complex Var. Elliptic Equ. 60 (2015), no. 11, 1505-1521.
[4] M. Allaoui, A. R. El Amrouss, A. Ourraoui, Existence results for a class of p(x)−Laplacian problems in R N . Computers & Mathematics with Applications 69(7): (2015) 582-591.
[5] M. Allaoui, A.R. El Amrouss, A. Ourraoui, Existence and multiplicity of solutions for a Steklov problem involving the p(x)-Laplacian operator, EJDE 132(2012) 1-12.
[6] M. Allaoui, A.R. El Amrouss, A. Ourraoui, Existence of infinitely many solutions for a Steklov problem involving the p(x)-Laplace operator, EJQTDE. 2014, No. 20, 1-10. [7] An. Lê, On the ?rst eigenvalue of the Steklov eigenvalue problem for the infinity Laplacian, EJDE Vol. 2006(2006), No. 111, 1-9. [8] A. Ayoujil, On the superlinear Steklov problem involving the p(x)-Laplacian, EJQTDE, 2014, No.38, 1-13.
[9] J.F. Bonder, J.D. Rossi, A nonlinear eigenvalue problem with indefinite weights related to the Sobolev trace embedding, Publicacions Matematiques, , (2002) 46:221-235.
[10] Y. Chen, S. Levine, R. Rao, Variable exponent, Linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006) 1383-1406.
[11] S.G. Deng, Eigenvalues of the p(x)-Laplacian Steklov problem J. Math. Anal. Appl. 339 (2008) 925-937.
[12] X.L. Fan, Global C 1, regularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 (2) (2007) 397-417.
[13] X.L. Fan , D.Zhao, On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263(2001) 424-446.
[14] X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces W m,p(x) (Ω), J. Math. Anal. Appl. 262(2001) 749-760.
[15] G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. anal. Appl. 73 (2010), 110-121.
[16] Juliano D.B. de Godo, Olímpio. H. Miyagaki, Rodrigo S. Rodrigues, On Steklov-Neumann boundary value problems for some quasilinear elliptic equations, Applied Mathematics Letters 45 (2015) 47-51.
[17] B. Karim, A. Zerouali and O. Chakrone, Existence and multiplicity of a-harmonic solutions for a Steklov problem with variable exponents, Bol. Soc. Paran. Mat., 2018, (3s)v. 32 2(2018), pp. 125-136.
[18] S. Kesavan, Topics in Functional Analysis and Applications, John Wiley & Sons, New York, 1989.
[19] O. Kovácik, J. R akosník; On spaces L p(x) and W k,p(x) , Czechoslovak Math. J. 41(1991) 592-618.
[20] A. Ourraoui, Multiplicity results for Steklov problem with variable exponent, Applied Mathematics and Computation 277(2016)34-43.
[21] M. Ruzicka, Electrorheological fluids: modeling and mathematical theory, 1748, Springer-Verlag, Berlin, 2000.
[22] O. Torné, Steklov problem with an indefinite weight for the p−Laplacian, Electron. J. Differential Equations 2005 (2005), no. 87, 8 pp.
[23] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Mathematics of the USSR-Izvestiya, vol. 9 (1987) 33-66.