Existence of Weak Solutions for a Nonlocal Problem Involving the p(x)-Laplace Operator

Abstract

This paper deals with the existence of weak solutions for some nonlocal problem involving the p (x)-Laplace operator. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions.

Existence and multiplicity of solutions for α(x)-Kirchhoff Equation with indefinite weights

“…We recall that such problems are proposed by Kirchhoff in [22] as an existence of the classical D'Alembert wave equations for free vibration of elastic strings. Note that elliptic and singular elliptic problems with α(x)-Laplace operator can be found in [1,2,3,4,5,8,12,13,15,17,18,19,20,21,24,25,26].…”

On critical variable-order Kirchhoff type problems with variable singular exponent

“…Readers who are interested to know the physical motivation behind the study of elliptic problems involving Kirchhoff operator can refer to Carrier [9]. In fact, there are only a few papers on the p(x)-Laplace operator involving singular nonlinearity and some of which can be found in the articles [2,17] and the references therein.…”

Solutions to a nonlocal elliptic problem in Orlicz-Sobolev spaces

“…Therefore, equation 1.1 particularly generalizes the problems involving variable exponent. This kind of equations have been intensively studied by many authors for the past two decades due to its significant role in many fields of mathematics, such as in the study of calculus of variations, partial differential equations [2,17,18], but also for their use in a variety of physical and engineering contexts: the modeling of electrorheological fluids [32], the analysis of Non-Newtonian fluids [36], fluid flow in porous media [3], magnetostatics [14], image restoration [11], and capillarity phenomena [8], see also, e.g., [4,5,6,7,9,12,13,16,23,35] and references therein. Therefore, equation (1.1) may represent a variety of mathematical models corresponding to certain phenomena:…”