Variable s(·)$$ s\left(\cdotp \right) $$‐order Kirchhoff‐type problem with a p(·)$$ p\left(\cdotp \right) $$‐fractional Laplace operator

Abstract

This paper deals with the existence and multiplicity of weak solutions for the following Kirchhoff‐type problem where models a Kirchhoff coefficient and is a variable ‐order ‐fractional Laplace operator, with and . Here, is a bounded smooth domain with for any . By using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle in two cases when the Carathéodory function having special structure, we establish conditions ensuring the multiplicity results.