Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T10:42:22.975Z Has data issue: false hasContentIssue false
  • English
  • Français

A p(x)-Laplacian extension of the Díaz-Saa inequality and some applications

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  24 January 2019

Peter Takáč  and
Jacques Giacomoni
Peter Takáč
Affiliation:
Institut für Mathematik, Universität Rostock Ulmenstraße 69, Haus 3 D-18055 Rostock, Germany (peter.takac@uni-rostock.de)
Jacques Giacomoni
Affiliation:
LMAP (UMR 5142) Université de Pau et des Pays de l'Adour Avenue de l'Université, F-64013 Pau cedex, France (jacques.giacomoni@univ-pau.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

The main result of this work is a new extension of the well-known inequality by Díaz and Saa which, in our case, involves an anisotropic operator, such as the p(x)-Laplacian, $\Delta _{p(x)}u\equiv {\rm div}( \vert \nabla u \vert ^{p(x)-2}\nabla u)$. Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions and comparison principles for some anisotropic quasilinear elliptic equations. Our proofs take advantage of certain convexity properties of the energy functional associated with the p(x)-Laplacian.

Keywords

p(x)-Laplacianquasilinear Dirichlet problem with variable exponentsray-strictly convex energy functionaluniqueness and comparison principles

MSC classification

Primary: 35J62: Quasilinear elliptic equations 35J92: Quasilinear elliptic equations with $p$-Laplacian
Secondary: 35B09: Positive solutions 35A02: Uniqueness problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 205 - 232
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Antontsev, S. N., Chipot, M. and Xie, Y.. Uniqueness results for equations of the p(x)-Laplacian type. Adv. Math. Sci. Appl. 17 (2007), 287–304.Google Scholar
2Antontsev, S. N. and Shmarev, S. I.. Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties. Nonlinear Anal. 65 (2006), 728–761.CrossRefGoogle Scholar
3Brézis, H. and Oswald, L.. Remarks on sublinear elliptic equations. Nonlinear Anal. 10 (1986), 55–64.CrossRefGoogle Scholar
4Chen, C.-Y., Kuo, Y.-C. and Wu, T.-F.. The Nehari manifold of a Kirchhoff type problem involving sign-changing weight functions. J. Diff. Equ. 250 (2011), 1876–1908.CrossRefGoogle Scholar
5Díaz, J. I. and Saa, J. E.. Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. Comptes Rendus Acad. Sc. Paris.Série I 305 (1987), 521–524.Google Scholar
6Diening, L., Harjulehto, P., Hästö, P. and Růžička, M.. Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, vol. 2017, (Berlin-Heidelberg: Springer-Verlag, 2011).CrossRefGoogle Scholar
7Fan, X.-L.. Global C 1, α regularity for variable exponent elliptic equations in divergence form. J. Diff. Equ. 235 (2007), 397–417.CrossRefGoogle Scholar
8Fan, X.-L.. Existence and uniqueness for the p(x)-Laplacian-Dirichlet problems. Math. Nachr. 284 (2011), 1435–1445.CrossRefGoogle Scholar
9Fan, X.-L. and Zhang, Q.-H.. Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear Anal. 52 (2003), 1843–1852.CrossRefGoogle Scholar
10Fan, X. and Zhao, D.. A class of De Giorgi type and H”older continuity. Nonlinear Anal. 36 (1999), 295–318.CrossRefGoogle Scholar
11Fan, X., Zhang, Q. and Zhao, D.. Eigenvalues of p(x)-Laplacian Dirichlet problems. J. Math. Anal. Appl. 302 (2005), 306–317.CrossRefGoogle Scholar
12Fleckinger, J., Hernández, J., Takáč, P. and de Thélin, F.. Uniqueness and positivity for solutions of equations with the p-Laplacian. In Proc. Conf. on Reaction-Diffusion Equations, 1995, Trieste, Italy (eds. Caristi, G. and Mitidieri, E.). Lecture Notes in Pure and Applied Mathematics, vol. 194, pp. 141–155 (New York and Basel: Marcel Dekker, 1998).Google Scholar
13Girg, P. and Takáč, P.. Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems. Ann. Inst. Henri Poincaré. Anal. Non Linéaire 9 (2008), 275–327.Google Scholar
14Krasnosel'skiĭ, M. A. and Zabreĭko, P. P.. Geometrical methods of nonlinear analysis (Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, 1984).CrossRefGoogle Scholar
15Mihăilescu, M. and Rădulescu, V.. On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent. Proc. Amer. Math. Soc. 135 (2007), 2929–2937.CrossRefGoogle Scholar
16Motreanu, V. V.. Uniqueness results for a Dirichlet problem with variable exponent. Comm. Pure. Appl. Anal. 9 (2010), 1399–1410. Online: doi: 10.3934/cpaa.2010.9.1399.CrossRefGoogle Scholar
17del Pino, M. A., Elgueta, M. and Manásevich, R. F.. A homotopic deformation along p of a Leray-Schauder degree result and existence for (|u′|p−2u′)′ + f(t, u) = 0, u(0) = u(T) = 0, p > 1. J. Diff. Equ. 80 (1989), 1–13.CrossRefGoogle Scholar
18Rădulescu, V. and Repovš, D.. Partial differential equations with variable exponents: variational methods and qualitative analysis (Boca Raton, FL: CRC Press, Taylor and Francis Group, 2015).CrossRefGoogle Scholar
19Růžička, M.. Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics, vol. 1748 (Berlin-Heidelberg: Springer-Verlag, 2000).CrossRefGoogle Scholar
20Struwe, M.. Variational methods, 2nd edn, in A Series of Modern Surveys in Mathematics, vol. 34 (Berlin-Heidelberg–New York: Springer-Verlag, 1996).CrossRefGoogle Scholar
21Takáč, P.. Nonlinear spectral problems for degenerate elliptic operators. In Handbook of differential equations: stationary partial differential equations (eds. Chipot, M. and Quittner, P.), vol. 1, pp. 385–489 (Amsterdam, The Netherlands: Elsevier Science B.V., 2004).CrossRefGoogle Scholar
22Takáč, P., Tello, L. and Ulm, M.. Variational problems with a p-homogeneous energy. Positivity 6 (2001), 75–94.CrossRefGoogle Scholar
23Zhang, Q.. A strong maximum principle for differential equations with nonstandard p(x)-growth conditions. J. Math. Anal. Appl. 312 (2005), 24–32.CrossRefGoogle Scholar