The Multiplicity of Nontrivial Solutions for a New $p\left(x\right)$-Kirchhoff-Type Elliptic Problem

Abstract

In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a p x -Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new p x -Kirchhoff problem has at least two nontrivial weak solutions.

A p(x)-Kirchhoff Type Problem Involving the p(x)-Laplacian-Like Operators With Dirichlet Boundary Condition

“…Perhaps the impulse for this comes from the new search field that reflects a new type of physical phenomenon is a class of nonlinear problems with variable exponents (see [26]). The motivation for this research comes from the application of similar models in physics to represent the behavior of elasticity [34] and electrorheological fluids (see [30,32]), which have the ability to modify their mechanical properties when exposed to an electric field (see [3,4,7,11,15,27,28,29]), specifically the phenomenon of capillarity, which depends on solid-liquid interfacial characteristics as surface tension, contact angle, and solid surface geometry.…”