Abstract
The p-Laplacian problem with the exponent of nonlinearity p depending on the solution u itself is considered in this work. Both situations when p(u) is a local quantity or when p(u) is nonlocal are studied here. For the associated boundary-value local problem, we prove the existence of weak solutions by using a singular perturbation technique. We also prove the existence of weak solutions to the nonlocal version of the associated boundary-value problem. The issue of uniqueness for these problems is addressed in this work as well, in particular by working out the uniqueness for a one dimensional local problem and by showing that the uniqueness is easily lost in the nonlocal problem.
Similar content being viewed by others
Change history
04 July 2019
In the Original Publication of the article, few errors have been identified in section 5 and acknowledgements section.
References
Andreianov, B., Bendahmane, M., Ouaro, S.: Structural stability for variable exponent elliptic problems. II. The \(p(u)\)-Laplacian and coupled problems. Nonlinear Anal. 72(12), 4649–4660 (2010)
Antontsev, S., Shmarev, S.: Evolution PDEs with Nonstandard Growth Conditions. Existence, Uniqueness, Localization, Blow-up. Atlantis Press, Paris (2015)
Blomgren, P., Chan, T., Mulet, P., Wong, C.: Total variation image restoration: Numerical methods and extensions. In: Proceedings of the IEEE International Conference on Image Processing, vol. 3, 384–387. IEEE Computer Society Press, Piscataway (1997)
Bollt, E., Chartrand, R., Esedoglu, S., Schultz, P., Vixie, K.: Graduated, adaptive image denoising: local compromise between total-variation and isotropic diffusion. Adv. Comput. Math. 31, 61–85 (2007)
Chipot, M.: Elements of Nonlinear Analysis. Birkhäuser, Basel (2000)
Chipot, M.: Elliptic Equations: An Introductory Course. Birkhäuser, Basel (2009)
Chipot, M., Chang, N.-H.: On some mixed boundary value problems with nonlocal diffusion. Adv. Math. Sci. Appl. 14(1), 1–24 (2004)
Chipot, M., Lovat, B.: Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems. Dyn. Contin. Discr. Impuls. Syst. Ser. A Math. Anal. 8(1), 35–51 (2001)
Chipot, M., Rodrigues, J.F.: On a class of nonlocal nonlinear elliptic problems. RAIRO: Modél. Math. Anal. Numér 26(3), 447–467 (1992)
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Birkhäuser/Springer, Heidelberg (2013)
Diening, L., Harjulehto, P., Hästo, P., Ru̇žička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Heidelberg (2011)
Glowinski, R., Marrocco, R.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires. RAIRO Ana. Numer. 9(R–2), 41–76 (1975)
Chan, T., Esedoglu, S., Park, F., Yip, A.: Total Variation Image Restoration: Overview and Recent Developments. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision, 17–32. Springer, New York (2006)
Türola, J.: Image denoising using directional adaptive variable exponents model. J. Math. Imaging. Vis. 57, 56–74 (2017)
Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 504(4), 675–710 (1986). English translation in Mah. USSR Izvestiya 29 (1987) 33–66
Zhikov, V.V.: On the technique for passing to the limit in nonlinear elliptic equations. Funct. Anal. Appl. 43(2), 96–112 (2009)
Zhikov, V.V.: On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J. Math. Sci. (N.Y.) 173(5), 463–570 (2011)
Acknowledgements
We are very grateful to the referees for their constructive remarks. This work was performed when the first author was visiting the USTC in Hefei and during a part time employment at the S. M. Nikolskii Mathematical Institute of RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, supported by the Ministry of Education and Science of the Russian Federation. He is grateful to these institutions for their support. Main part of this work was carried out also during the visit of the second author to the University of Zurich during the first quarter of 2018. Besides the Grant SFRH/BSAB/135242/2017 of the Portuguese Foundation for Science and Technology (FCT), Portugal, which made this visit possible, the second author also wishes to thank to Prof. Michel Chipot who kindly welcomed him at the University of Zurich.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Giga.
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
Chipot, M., de Oliveira, H.B. Some results on the p(u)-Laplacian problem. Math. Ann. 375, 283–306 (2019). https://doi.org/10.1007/s00208-019-01803-w
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-019-01803-w