EXISTENCE OF WEAK SOLUTIONS FOR $p(x)$-KIRCHHOFF-TYPE EQUATION

Abstract

This paper is considerned with the existence of solutions for p(x)-Kirchhoff-type problem under with Dirichlet boundary condition. By direct variational method and the Mountain Pass theorem, we establish some conditions that ensure the existence nontrivial weak solutions for the problem.

Existence and multiplicity of solutions for a p(x)-Kirchhoff type equation with Steklov boundary

“…In order to discuss problem (1.1), we need some definitions and basic properties of variable exponent Lebesgue-Sobolev spaces which will be used later [16,22,28].…”

“…This equation is an extension of the classical D'Alambert's wave equation, by considering the effects of the changes in the length of the strings during the vibrations. For p (x)-Kirchhoff type problems see, for example, [10,11,28]. Differential and partial differential problems with variable exponent growth condition have been received considerable attention in recent years.…”

Weak solutions for discrete Kirchhoff type equations with Dirichlet boundary conditions

“…After that, several physicists also considered such equations for their researches in the theory of nonlinear vibrations theoreticallly or experimentally [10,11,32,33]. As far as we know, the first study which deals with anisotropic discrete boundary value problems of p(•)-Kirchhoff type difference equation was done by Yucedag (see [36]). A more general study of the problem of Yucedag has been done by Koné et al (see [25]).…”