Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

A class of De Giorgi type and local boundedness
  • Home
  • /
  • A class of De Giorgi type and local boundedness
  1. Home /
  2. Archives /
  3. Vol 51, No 2 (June 2018) /
  4. Articles

A class of De Giorgi type and local boundedness

Authors

  • Duchao Liu
  • Jinghua Yao

Keywords

Musielak-Sobolev space, local bounded property

Abstract

Under appropriate assumptions on the $N(\Omega)$-function, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev spaces. As the applications, the local boundedness property of the minimizers for a class of the energy functionals in Musielak-Orlicz-Sobolev spaces is proved; and furthermore, the local boundedness of the weak solutions for a class of fully nonlinear elliptic equations is provided.

References

E. Acerbi and N. Fusco, Regularity for minimizers of non-quadratic functionals: The case 1 < p < 2, J. Math. Anal. Appl. 140 (1989), 115–135.

T. Adamowicz and O. Toivanen, Hölder continuity of quasiminimizers with nonstandard growth, Nonlinear Anal. 125 (2015), 433–456.

R. Adams, Sobolev Spaces, Academic Press, New York, 1975.

A. Benkirane and M. Sidi El Vally, An existence result for nonlinear elliptic equations in Musielak–Orlicz–Sobolev spaces, Bull. Belg. Math. Soc. 20 (2013), 1–187.

S. Byun, J. Ok and L. Wang, W 1,p( · ) -regularity for elliptic eqautions with measurable coefficients in nonsmooth domains, Comm. Math. Phys. 329 (2014), 937–958.

F. Cammaroto and L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator, Nonlinear Anal. 74 (2011), 1841–1852.

L. Diening, P. Hästö and S. Roudenko, Function spaces of variable smoothness and integrability, J. Func. Anal. 256 (2009), 1731–1768.

L. Diening, B. Stroffolini and A. Verde, Everywhere regularity of functionals with ϕ-growth, Manuscripta Math. 129 (2009), 449–481.

L. Diening, B. Stroffolini and A. Verde, Lipschitz regularity for some asymptotically convex problems, ESAIM Control Optim. Calc. Var. 17 (2011), 178–189.

T.K. Donaldson and N.S. Trudinger, Orlicz–Sobolev spaces and imbedding theorems, J. Func. Anal. 8 (1971), 52–75.

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

X.L. Fan, Differential equations of divergence form in Musielak–Sobolev spaces and subsupersolution method, J. Math. Anal. Appl. 386 (2012), 593–604.

X.L. Fan, An imbedding theorem for Musielak–Sobolev spaces, Nonlinear Anal. 75 (2012), 1959–1971.

X. Fan and C. Guan, Uniform convexity of Musielak–Orlicz–Sobolev spaces and applications, Nonlinear Anal. 73 (2010), 163–175.

X. Fan and D. Zhao, A class of de giorgi type and Hölder continuous, Nonlinear Anal. 36 (1999), 295–318.

X.L. Fan and D. Zhao, On the generalized Orlicz–Sobolev space W k,p(x) (Ω), J. Gansu Educ. College 12 (1998), 1–6.

M. Garcı́a-Huidobro, V.K. Le, R. Manásevich and K. Schmitt, On principle eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting, Nonlinear Differential Equations Appl. 6 (1999), 207–225.

P. Harjulehto, P. Hästö and O. Toivanen, Hölder regularity of quasiminimizers under generalized growth conditions, Calc. Var. 56 (2017), no. 22.

D. Liu, B. Wang and P. Zhao, On the trace regularity results of Musielak–Orlicz–Sbolev spaces in a bounded domain, Comm. Pure Appl. Anal. 15 (2016), 1643–1659.

D. Liu and P. Zhao, Solutions for a quasilinear elliptic equation in Musielak–Sobolev spaces, Nonlinear Anal. Real World Appl. 26 (2016), 315–329.

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer, Berlin, 1983.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2018-04-22

How to Cite

1.
LIU, Duchao and YAO, Jinghua. A class of De Giorgi type and local boundedness. Topological Methods in Nonlinear Analysis. Online. 22 April 2018. Vol. 51, no. 2, pp. 345 - 370. [Accessed 3 July 2025].
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 51, No 2 (June 2018)

Section

Articles

Stats

Number of views and downloads: 0
Number of citations: 0

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop