Infinitely many solutions for anisotropic elliptic equations with variable exponent
Keywords:Quasilinear elliptic equations, variable exponent Lebesgue space, anisotropic space, Fountain theorem, dual Fountain theorem
In this article, we study the existence and multiplicity of solutions for a class of anisotropic elliptic equations
First we establisch that anisotropic space is separable and by using the Fountain theorem, and dual Fountain theorem we prove, under suitable conditions, that the problem (P) admits two sequences of weak solutions.
G. A. Afrouzi, M. Mirzapour, and V. D. Rădulescu, “Qualitative Properties of Anisotropic Elliptic Schrödinger Equations”, Advanced Nonlinear Studies, vol. 14, pp. 719-736, 2014.
C. O. Alves, and J. L. P. Barreiro, “Existence and multiplicity of solutions for a p(x)- Laplacian equation with critical growth”, Journal of Mathematical Analysis and Applications, vol. 403, pp. 143-154, 2013.
A. Ambrosetti, and P. H. Rabinowitz, “Dual variational methods in critical points theory and applications”, Journal of Functional Analysis, vol. 14, pp. 349-381, 1973.
T. Bartsch, “Infinitely many solutions of a symmetric Dirichlet problema”, Nonlinear Analysis, vol. 20, pp. 1205-1216, 1993.
M.M. Boureanu, “Infinitely many solution for a class of degenerate anisotropic elliptic problems with variable exponent”, Taiwanese journal of mathematics, vol. 15, pp. 2291-2310, 2011.
M. M. Boureanu, P. Pucci, and V. D. Rădulescu, “Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent”, Complex Variables and Elliptic Equations, vol. 56, pp. 755-767, 2011.
H. Brezis, Analyse Fonctionnelle. Théorie, Méthodes et Applications. París: Masson, 1992.
D. E. Edmunds, J. Rákosník, “Sobolev embedding with variable exponent”, Studia Mathematica, vol. 143, pp. 267-293, 2000.
X. L. Fan, “Anisotropic variable exponent Sobolev spaces and ⃗p (x)−Laplacian equations", Complex Variables and Elliptic Equations, vol. 56, no. 7-9, pp. 623-642, 2011.
X. L. Fan, “Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients”, Journal of Mathematical Analysis and Applications, vol. 312, pp. 464-477, 2005.
X. L. Fan, and X. Y. Han, “Existence and multiplicity of solutions for p(x)− Laplacian equations in RN”, Nonlinear analysis, vol. 59, pp. 173-188, 2004.
X. L. Fan, J. S. Shen, and D. Zhao, “Sobolev embedding theorems for spaces Wk,p(x)” , Journal of Mathematical Analysis and Applications, vol. 262, pp. 749-760, 2001.
X.L. Fan, and D. Zhao, “On the spaces Lp(x) and Wm,p(x)”, Journal of Mathematical Analysis and Applications, vol. 263, pp. 424-446, 2001.
B. Kone, S. Ouaro, and S. Traore, “Weak solutions for anisotropic nonlinear elliptic equations with variable exponents”, Electronic Journal of Differential Equations, vol. 2009, no. 144. pp. 1-11, 2009.
M. Mihailescu, and G. Morosanu, “Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions”, Applicable Analysis, vol. 89, no. 2, pp. 257-271, 2010.
M. Mihailescu, P. Pucci, and V. Radulescu, “Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent”, Journal of Mathematical Analysis and Applications, vol. 340, pp. 687-698, 2008.
T. G. Myers, “Films with high surface tensión”, SIAM Review, vol. 40, no. 3, pp. 441-462, 1998.
H. Nguyen, and K. Schmitt, “Nonlinear elliptic Dirichlet and no-flux boundary value problems”, Annals of the University of Bucharest (Mathematical Series), vol. 61, no. 3, pp. 201-217, 2012.
M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory. Berlin: Springer, 2000.
M. Willem, Minimax Theorems. Boston: Birkhauser, 1996.
J.F. Zhao, Structure Theory of Banach Spaces. Wuhan: Wuhan University Press, 1991.
L. Zhao, P. Zhao, and X. Xie, “Existence and multiplicity of solutions for divergence type elliptic equations”, Electronic Journal of Differential Equations, no. 43, pp. 1-9, 2011.
V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Berlin: Springer, 1994.
How to Cite
Copyright (c) 2021 Abdelrachid El Amrouss, Ali El Mahraoui
This work is licensed under a Creative Commons Attribution 4.0 International License.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.