Eigenvalues of p(x)-Laplacian Dirichlet problem

Abstract

This paper studies the eigenvalues of the p(x)-Laplacian Dirichlet problemwhere Ω is a bounded domain in R N and p(x) is a continuous function onΩ such that p(x) > 1. We show that Λ, the set of eigenvalues, is a nonempty infinite set such that sup Λ = +∞. We present some sufficient conditions for inf Λ = 0 and for inf Λ > 0, respectively. 

On an Eigenvalue Problem for an Anisotropic Elliptic Equation Involving Variable Exponents

“…On the other hand, we point out that our study extends to the case of anisotropic equations the results obtained in [22] and generalizes some other existing results on eigenvalue problems involving variable exponent growth conditions [11,12,20,21,23]. Finally, we note that equations of type (1) are models for various phenomena which arise from the study of electrorheological fluids (see [7,14,20,29,30]), image processing (see [6]), or the theory of elasticity (see [35]).…”

Existence of solutions for an elliptic p(x)-Kirchhoff-type systems in unbounded domain

“…More precisely, Kirchhoff proposed the following model 2) have the following meanings: E is the Young modulus of the material, ρ is the mass density, L is the length of the string, h is the area of cross-section, and P 0 is the initial tension. The study of elliptic problems involving p(x)−Laplacian has interested in recent years, for the existence of solutions see [1], [9] and [12], and the eigenvalue involving p (x) −Laplacian problems see [10] and [11].…”

Form estimates for the 𝑝(𝑥)-Laplacean

“…If α = 0, L is termed the p(x)-Laplacean. These equations arise in such fields as electrorheological fluids, and the interested readers may find results and applications for such problems, as well as properties of the associated L p(x) and W k,p(x) spaces in the articles [1], [2], [5], [7], [8], [9], [10], [12] and the references therein.…”

“…[7]. In this reference the authors consider the case of Dirichlet boundary conditions in a domain Ω and show the existence of infinitely many eigenvalues.…”