Abstract
The existence of nontrivial weak solutions of the anisotropic problem with variable exponent and Robin boundary conditions is the focus of this research. We obtain the existence of solutions using the Mountain pass theorem and the Ekeland variational principle as well as the theory of the anisotropic variable exponent Sobolev space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Acerbi and G. Mingione, “Gradient estimates for the \(p(x)\)-Laplacian system,” J. Reine Angew. Math. 584, 117–148 (2005).
T. C. Halsey, “Electrorheological fluids,” Sci. 5083 (258), 761–766 (1992).
V. Zhikov, “Averaging of functionals in the calculus of variations and elasticity,” Math. USSR Izv. 29, 33–66 (1987).
L. Diening, “Theoretical and numerical results for electrorheological fluids,” Ph. D thesis, University of Freiburg, Germany (2002).
Y. Chen, S. Levine, and R. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM J. Appl. Math. 66 (4), 1383–1406 (2006).
S. Antontsev and S. Shmarev, “A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions,” Nonlinear Anal. 60 (3), 515–545 (2005).
M. Mihailescu, P. Pucci and V. Radulescu, “Nonhomogeneous boundary value problems in anisotropic Sobolev spaces,” C. R. Math. 345, 561–566 (2007).
B. Ellahyani and A. El Hachimi, “Existence and multiplicity of solutions for anisotropic elliptic problems with variable exponent and nonlinear Robin conditions,” Electron. J. Differ. Equ. 188, 1–17 (2017).
M. Hsini, N. Irzi, and K. Kefi, “Existence of solutions for a \(p(x)\)-biharmonic problem under Neumann boundary conditions,” Appl. Anal. 100 (10), 2188–2199 (2021).
J. Musielak, in Orlicz spaces and Modular Spaces, Lecture Notes in Mathematics (Springer, Berlin, 1983), Vol. 1034.
O. Kováčik and J. Rákosník, “On the spaces \(L^{p(x)}(\Omega)\) and \(W^{1,p(x)}(\Omega)\),” Czechoslovak Math. J. 41, 592–618 (1991).
X. L. Fan and D. Zhao, “On the spaces \(L^{p(x)}\) and \(W^{m,p(x)}\),” J. Math. Anal. Appl. 263, 424–446 (2001).
X. L. Fan and Q. H. Zhang, “Existence of solutions for \(p(x)\)-Laplacian Dirichlet problems,” Nonlinear Anal. 52, 1843–1852 (2003).
X. L. Fan, J. S. Shen, and D. Zhao, “Sobolev embedding theorems for spaces \(W^{k,p(x)}(\Omega)\),” J. Math. Anal. Appl. 262, 749–760 (2001).
I. Ekeland, “On the variational principle,” J. Math. Anal. Appl. 47, 324–353 (1974).
X. L. Fan and X. Y. Han, “Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\),” Nonlinear Anal. 59, 173–188 (2004).
X. L. Fan, “On nonlocal \(p(x)\)-Laplacian Dirichlet problems,” Nonlinear Anal. 72, 3314–3323 (2010).
S. Antontsev and S. Shmarev, “Anisotropic parabolic equations with variable nonlinearity,” Publ. Mat. 53 (2), 355–399 (2009).
S. Antontsev and S. Shmarev, “Evolution PDEs with nonstandard growth conditions,” Atlantis Stud. Differ. Equ. 4 (2015).
M. M. Boureanu, “Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent,” Taiwanese J. Math. 15 (5), 2291–2310 (2011).
M. M. Boureanu and V. Radulescu, “Anisotropic Neumann problems in Sobolev spaces with variable exponent,” Nonlinear Anal. 75 (12), 4471–4482 (2012).
B. Kone, S. Ouaro, and S. Traore, “Weak solutions for anisotropic nonlinear elliptic equations with variable exponents,” Electron. J. Differ. Equ. 144, 1–11 (2009).
Acknowledgments
We thank the referees for help to revise this paper and for important remarks.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hsini, M., Mbarki, L. & Das, K. Existence of Solutions of Anisotropic Problems with Variable Exponents with Robin Boundary Conditions. Math Notes 112, 898–910 (2022). https://doi.org/10.1134/S0001434622110244
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434622110244