Abstract
This paper is concerned with the existence of "weak solution" for a Dirichlet boundary value problems involving the p(x)-Laplacian operator depending on three real parameters. The proof of the main result is constructed by utilizing the topological degree for a class of demicontinuous operators of generalized \((S_{+})\) type and the theory of variable-exponent Sobolev spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aboulaich, R., Meskine, D., Souissi, A.: New diffusion models in image processing. Comput. Math. Appl. 56, 874–882 (2008)
Acerbi, E., Mingione, G.: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001)
Alsaedi, R.: Perturbed subcritical Dirichlet problems with variable exponents. Elec. J. Differ. Equ. 295, 1–12 (2016)
Antontsev, S., Shmarev, S.: A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions. Nonlinear Anal. 60, 515–545 (2005)
Berkovits, J.: Extension of the Leray-Schauder degree for abstract Hammerstein type mappings. J Differ. Equ. 234, 289–310 (2007)
Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)
El Ouaarabi, M., Allalou, C., Melliani, S.: Existence result for Neumann problems with p(x)-Laplacian-like operators in generalized Sobolev spaces. Rend. Circ. Mat. Palermo, II. Ser.
Fan, X.L., Han, X.Y.: Existence and multiplicity of solutions for \(p(x)-\)Laplacian equations in \({\mathbb{R}}^N\). Nonlinear Anal. 59, 173–188 (2004)
Fan, X.L., Zhang, Q.H.: Existence of solutions for \(p(x)\)-laplacian Dirichlet problem. Nonlinear Anal. 52, 1843–1852 (2003)
Fan, X.L., Zhao, D.: On the Spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)
Guo, Y., Ye, G.: Existence and uniqueness of weak solutions to variable-order fractional Laplacian equations with variable exponents. J. Funct. Spaces (2021). https://doi.org/10.1155/2021/6686213
Harjulehto, P., Hästö, P., Koskenoja, M., Varonen, S.: The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values. Potential Anal. 3(25), 205–222 (2006)
Henriques, E., Urbano, J.M.: Intrinsic scaling for PDEs with an exponential nonlinearity. Indiana Univ. Math. J. 55, 1701–1722 (2006)
Kim, I.S., Hong, S.J.: A topological degree for operators of generalized \((S_{+})\) type. Fixed Point Theory Appl. 1, 1–16 (2015)
Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{1, p(x)}\). Czechoslov. Math. J. 41(4), 592–618 (1991)
Marcellini, P.: Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions. Arch. Ration. Mech. Anal. 105, 267–284 (1989)
Ouaarabi, M.E., Abbassi, A., Allalou, C.: Existence result for a Dirichlet problem governed by nonlinear degenerate elliptic equation in weighted Sobolev spaces. J. Elliptic Parabol. Equ. 7(1), 221–242 (2021)
Ouaarabi, M.E., Allalou, C., Abbassi, A.: On the Dirichlet problem for some nonlinear degenerated elliptic equations with weight. In:7th International Conference on Optimization and Applications (ICOA), pp. 1–6 (2021)
Ouaarabi, M.E., Abbassi, A., Allalou, C.: Existence result for a general nonlinear degenerate elliptic problems with measure datum in weighted Sobolev spaces. Int. J. Optim. Appl. 1(2), 1–9 (2021)
Ouaarabi, M.E., Abbassi, A., Allalou, C.: Existence and uniqueness of weak solution in weighted Sobolev spaces for a class of nonlinear degenerate elliptic problems with measure data. Int. J. Nonlinear Anal. Appl. 13(1), 2635–2653 (2021)
Ragusa, M.A.: On weak solutions of ultraparabolic equations. Nonlinear Anal. 47(1), 503–511 (2001)
Rajagopal, K.R., Ruzicka, M.: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13(1), 59–78 (2001)
Ruzicka, M.: Electrorheological fuids: Modeling and mathematical theory. Springer Science and Business Media, Berlin (2000)
Samko, S.G.: Density of \(C_{0}^{\infty }({\mathbb{R}}^{N})\) in the generalized Sobolev spaces \(W^{m, p(x)}({\mathbb{R}}^{N})\). Doklady Mathematics. 60(3), 382–385 (1999)
Zeidler, E.: Nonlinear functional analysis and its applications II/B. Springer-Verlag, New York (1990)
Zhao, D., Qiang, W.J., Fan, X.L.: On generalizerd Orlicz spaces \(L^{p(x)}(\Omega )\). J. Gansu Sci. 9(2), 1–7 (1996)
Zhikov, V.V.E.: Averaging of functionals of the calculus of variations and elasticity theory. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 50(4), 675–710 (1986)
Acknowledgements
The authors would like to thank the anonymous referees and the editor for their careful reading and valuable comments and suggestions on this article.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
El Ouaarabi, M., Allalou, C. & Melliani, S. Existence of weak solution for a class of p(x)-Laplacian problems depending on three real parameters with Dirichlet condition. Bol. Soc. Mat. Mex. 28, 31 (2022). https://doi.org/10.1007/s40590-022-00427-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40590-022-00427-6
Keywords
- p(x)-Laplacian operator
- Dirichlet boundary value problems
- Hammerstein equation
- Variable-exponent Sobolev spaces
- Topological degree methods