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Volume 6, Issue 4

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

Ky Ho
/ Inbo Sim
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.

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.

• [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.

• [5]

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

• [6]

X. Fan, J. S. Shen and D. Zhao, Sobolev embedding theorems for spaces ${W}^{k,p\left(x\right)}\left(\mathrm{\Omega }\right)$, 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.

• [8]

X. Fan and D. Zhao, On the spaces ${L}^{p\left(x\right)}\left(\mathrm{\Omega }\right)$ and ${W}^{m,p\left(x\right)}\left(\mathrm{\Omega }\right)$, J. Math. Anal. Appl. 263 (2001), 424–446.

• [9]

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

• [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.

• [11]

K. Ho and I. Sim, Existence results for degenerate $p\left(x\right)$-Laplace equations with Leray–Lions type operators, Sci. China Math., to appear.

• [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.

• [14]

O. Kovăčik and J. Răkosnik, On spaces ${L}^{p\left(x\right)}$ and ${W}^{k,p\left(x\right)}$, 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.

• [16]

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

• [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.

• [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 $p\left(x\right)$-Laplacian operator, Adv. Nonlinear Anal. 4 (2015), no. 4, 285–293.

• [21]

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

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,

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