On a nonhomogeneous quasilinear eigenvalue problem in sobolev spaces with variable exponent

By M. Mihăilescu, Vicentiu D. Rădulescu. 2006

284 citations recorded by the source corpus.

We consider the nonlinear eigenvalue problem -div (|∇ u | p(x) - 2 ∇ u ) = λ( u ( q(x) - 2 u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R N with smooth boundary and p, q are continuous functions on Ω such that 1 0 sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/4388d6f6c35e820682b9c114b33f3d58fc8e2120

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In Applied Mathematics

Citations in the corpus

On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting (2016, result): “We refer especially to the results in [13, 18, 20, 12, 21, 22, 23].”
p(x)-Laplacian with indefinite weight (2011, methodology): “We also refer to the recent papers [2, 4, 5, 9, 11, 12, 17, 18, 19] for the treatment of nonlinear boundary value problems in Lebesque-Sobolev spaces with variable exponent.”
The concentration-compactness principle for variable exponent spaces and applications (2009, unknown): “See, for instance [5, 6, 8, 13, 14] and references therein.”
Overview of differential equations with non-standard growth (2010, background): “[153] Mihăilescu, M.; Pucci, P.; Rădulescu, V.: Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent.”
Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent (2008, background): “[14,15], Mihăilescu and Rădulescu [22–24], Rajagopal and Ruzicka [32], and Ruzicka [33].”
Local-to-global results in variable exponent spaces (2009, unknown): “[3, 4, 6, 17, 19, 22, 24, 31, 39].”
Eigenvalue problems for anisotropic discrete boundary value problems (2009, background): “Problems like (2) have been intensively studied in the last decades since they can model various phenomena arising from the study of elastic mechanics (see, Zhikov [24]), electrorheological fluids (see, Acerbi and Mingione [1], Diening [6], Halsey [9], Ruzicka [18], Mihăilescu and Rădulescu [12, 13,”
Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces (2012, background): “The interest in analysing this kind of problems is motivated by some recent advances in the study of eigenvalue problems involving non-homogeneous operators in the divergence form; see, for instance, the papers [1–12].”
Solutions of nonlinear problems involving p(x)-Laplacian operator (2015, background): “Moreover, the nonlinear problems involving p(x)-Laplacian operator have become extremely attractive over the last two decades [3, 7, 15, 19].”
On a p(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p(x)}$$\end{document}-biharmonic problem with singula (2017, methodology): “This approach was introduced for the first time in [23] in the framework of problems with variable exponent.”
Existence and multiplicity results for a newp(x)-Kirchhoff problem (2019, background): “Some important and interesting results can be found, for example, in [4, 5, 3, 2, 7, 10, 8, 11, 12, 13, 15, 18, 21, 19, 22, 23, 24, 25, 29, 30, 31, 32, 33, 40] and references therein.”