Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco

4 Issues per year


IMPACT FACTOR 2016: 2.844

CiteScore 2016: 2.41

SCImago Journal Rank (SJR) 2016: 1.951
Source Normalized Impact per Paper (SNIP) 2016: 2.187

Mathematical Citation Quotient (MCQ) 2016: 1.96

Online
ISSN
2191-950X
See all formats and pricing
More options …

A-priori bounds and existence for solutions of weighted elliptic equations with a convection term

Ky Ho / Inbo SimORCID iD: http://orcid.org/0000-0002-1618-054X
Published Online: 2016-04-07 | DOI: https://doi.org/10.1515/anona-2015-0177

Abstract

We investigate weighted elliptic equations containing a convection term with variable exponents that are subject to Dirichlet or Neumann boundary condition. By employing the De Giorgi iteration and a localization method, we give a-priori bounds for solutions to these problems. The existence of solutions is also established using Brezis’ theorem for pseudomonotone operators.

Keywords: weighted variable exponent Lebesgue–Sobolev spaces; a-priori bound; De Giorgi iteration; localization method; pseudomonotone operators

MSC 2010: 35J20; 35J60; 35J70; 47J10; 46E35

References

  • [1]

    M. B. Benboubker, E. Azroul and A. Barbara, Quasilinear elliptic problems with nonstandard growth, Electron. J. Differential Equations 62 (2011), 1–16. Google Scholar

  • [2]

    M.-M. Boureanu, A new class of nonhomogeneous differential operator and applications to anisotropic systems, Complex Var. Elliptic Equ. (2015), 10.1080/17476933.2015.1114614. Google Scholar

  • [3]

    M.-M. Boureanu and V. Rădulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal. 75 (2012), 4471–4482. Web of ScienceCrossrefGoogle Scholar

  • [4]

    L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math. 2017, Springer, Berlin, 2011. Web of ScienceGoogle Scholar

  • [5]

    X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl. 339 (2008), 1395–1412. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    X. Fan, J. S. Shen and D. Zhao, Sobolev embedding theorems for spaces Wk,p(x)(Ω), J. Math. Anal. Appl. 262 (2001), 749–760. Google Scholar

  • [7]

    X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal. 36 (1999), 295–318. CrossrefGoogle Scholar

  • [8]

    X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446. Web of ScienceGoogle Scholar

  • [9]

    L. Gasiński and N. S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations 42 (2011), 323–354. Web of ScienceCrossrefGoogle Scholar

  • [10]

    K. Ho and I. Sim, Existence and some properties of solutions for degenerate elliptic equations with exponent variable, Nonlinear Anal. 98 (2014), 146–164; corrigendum, Nonlinear Anal. 128 (2015), 423–426. Web of ScienceCrossrefGoogle Scholar

  • [11]

    K. Ho and I. Sim, Existence results for degenerate p(x)-Laplace equations with Leray–Lions type operators, Sci. China Math., to appear. Web of ScienceGoogle Scholar

  • [12]

    Y. H. Kim and I. Sim, Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents, Discrete Contin. Dyn. Syst. 2013 (2013), 695–707. Google Scholar

  • [13]

    Y. H. Kim, L. Wang and C. Zhang, Global bifurcation of a class of degenerate elliptic equations with variable exponents, J. Math. Anal. Appl. 371 (2010), 624–637. Web of ScienceCrossrefGoogle Scholar

  • [14]

    O. Kovăčik and J. Răkosnik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41 (1991), no. 116, 592–618. Google Scholar

  • [15]

    M. Mihăilescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A 462 (2006), 2625–2641. CrossrefGoogle Scholar

  • [16]

    V. Rădulescu, Nonlinear elliptic equations with variable exponent: Old and new, Nonlinear Anal. 121 (2015), 336–369. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    V. Rădulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Boca Raton, 2015. Google Scholar

  • [18]

    D. Repovš, Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. 13 (2015), no. 6, 645–661. CrossrefGoogle Scholar

  • [19]

    P. Winkert and R. Zacher, A priori bounds for weak solutions to elliptic equations with nonstandard growth, Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 4, 865–878; corrigendum, http://www.math.winkert.de/publications/final/FinalDCDS-S2015.pdf. Web of Science

  • [20]

    Z. Yücedağ, Solutions of nonlinear problems involving p(x)-Laplacian operator, Adv. Nonlinear Anal. 4 (2015), no. 4, 285–293. Web of ScienceGoogle Scholar

  • [21]

    E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B: Nonlinear Monotone Operators, Springer, New York, 1990. Google Scholar

About the article

Received: 2015-12-22

Revised: 2016-03-03

Accepted: 2016-03-04

Published Online: 2016-04-07

Published in Print: 2017-11-01


The first author was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports. The second author was supported by the National Research Foundation of Korea Grant funded by the Korea Government (MEST) (NRF-2015R1D1A3A01019789).


Citation Information: Advances in Nonlinear Analysis, Volume 6, Issue 4, Pages 427–445, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2015-0177.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in