Opuscula Math. 38, no. 6 (2018), 765-777

Opuscula Mathematica

Improved bounds for solutions of ϕ-Laplacians

Waldo Arriagada
Jorge Huentutripay

Abstract. In this short paper we prove a parametric version of the Harnack inequality for \(\phi\)-Laplacian equations. In this sense, the estimates are optimal and represent an improvement of previous bounds for this kind of operators.

Keywords: Orlicz-Sobolev space, Harnack inequality, \(\phi\)-Laplacian.

Mathematics Subject Classification: 35B50, 35J20, 35J60.

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  1. R.A. Adams, John J.J.F. Fournier, Sobolev Spaces, vol. 140, Pure and Applied Mathematics, 2nd ed., Elsevier B.V., Amsterdam, 2003.
  2. W. Arriagada, J. Huentutripay, A Harnack's inequality in Orlicz-Sobolev spaces, Studia Math. 243 (2018), 117-137.
  3. W. Arriagada, J. Huentutripay, Regularity, positivity and asymptotic vanishing of solutions of a \(\phi\)-Laplacian, Anal. Ştiinţ Univ. "Ovidius" Constanţa Ser. Mat. 25 (2017) 3, 59-72.
  4. H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983.
  5. E. DiBenedetto, Partial Differential Equations, Cornerstones, Birkhäuser Boston, Inc., Boston, MA, 2nd ed., 2010.
  6. M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic System, Princeton Univ. Press, Princeton, New Jersey, 1983.
  7. D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Germany, reprint of the 1998 edition, 2001.
  8. J.P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, [in:] S. Fučík, A. Kufner (eds.), Nonlinear Analysis, Function Spaces and Applications, Proceedings of a Spring School held in Horní Bradlo, 1978, vol. 1, BSB B.G. Teubner Verlagsgesellschaft, Leipzig, 1979. Teubner Texte zur Mathematik, 59-94.
  9. U. Kaufmann, I. Medri, One-dimensional singular problems involving the \(p\)-Laplacian and nonlinearities indefinite in sign, Adv. Nonlinear Anal. 5 (2016), 251-259.
  10. M. Krasnosel'skii, J. Rutickii, Convex Functions and Orlicz Space, English translation P. Noordhoff Ltd., Groningen, 1961.
  11. G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991) 2-3, 311-361.
  12. B. Maultsby, Uniqueness of solutions to singular \(p\)-Laplacian equations with subcritical nonlinearity, Adv. Nonlinear Anal. 6 (2017) 1, 37-59.
  13. J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591.
  14. V. Rădulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal. 121 (2015), 336-369.
  15. V. Rădulescu, D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Chapman & Hall/CRC Monographs and Research Notes in Mathematics, Taylor & Francis Group, Boca Raton FL, 2015.
  16. I.-L. Stăncuţ, I.D. Stîrcu, Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces, Opuscula Math. 36 (2016) 1, 81-101.
  • Waldo Arriagada
  • Khalifa University, Department of Applied Mathematics and Sciences, P.O. Box 127788, Abu Dhabi, United Arab Emirates
  • Jorge Huentutripay
  • Universidad Austral de Chile, Instituto de Ciencias Físicas y Matemáticas, Campus Isla Teja, Valdivia, Chile
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2018-01-11.
  • Revised: 2018-02-21.
  • Accepted: 2018-02-21.
  • Published online: 2018-07-05.
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Cite this article as:
Waldo Arriagada, Jorge Huentutripay, Improved bounds for solutions of ϕ-Laplacians, Opuscula Math. 38, no. 6 (2018), 765-777, https://doi.org/10.7494/OpMath.2018.38.6.765

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