Opuscula Mathematica
Opuscula Math. 32, no. 4 (), 731-750
http://dx.doi.org/10.7494/OpMath.2012.32.4.731
Opuscula Mathematica

On a class of nonhomogenous quasilinear problems in Orlicz-Sobolev spaces


Abstract. We study the nonlinear boundary value problem \(-div ((a_1(|\nabla u(x)|)+a_2(|\nabla u(x)|))\nabla u(x))=\lambda |u|^{q(x)-2}u-\mu |u|^{\alpha(x)-2}u\) in \(\Omega\), \(u = 0\) on \(\partial \Omega\) , where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary, \(\lambda\), \(\mu\) are positive real numbers, \(q\) and \(\alpha\) are continuous functions and \(a_1\), \(a_2\) are two mappings such that \(a_1(|t|)t\), \(a_2(|t|)t\) are increasing homeomorphisms from \(\mathbb{R}\) to \(\mathbb{R}\). The problem is analyzed in the context of Orlicz-Soboev spaces. First we show the existence of infinitely many weak solutions for any \(\lambda,\mu \gt 0\). Second we prove that for any \(\mu \gt 0\), there exists \(\lambda_*\) sufficiently small, and \(\lambda ^*\) large enough such that for any \(\lambda \in (0,\lambda_*)\cup(\lambda^*,\infty)\), the above nonhomogeneous quasilinear problem has a non-trivial weak solution.
Keywords: variable exponent Lebesgue space, Orlicz-Sobolev space, critical point, weak solution.
Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02.
Cite this article as:
Asma Karoui Souayah, On a class of nonhomogenous quasilinear problems in Orlicz-Sobolev spaces, Opuscula Math. 32, no. 4 (2012), 731-750, http://dx.doi.org/10.7494/OpMath.2012.32.4.731
 
Download this article's citation as:
a .bib file (BibTeX), a .ris file (RefMan), a .enw file (EndNote)
or export to RefWorks.

RSS Feed

horizontal rule

ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
Copyright © 2003−2017 OPUSCULA MATHEMATICA
Contact: opuscula@agh.edu.pl
Made by Tomasz Zabawa

horizontal rule

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.