Three solutions for a double phase variable exponent Kirchhoff problem

In this article, we study a double phase variable exponent Kirchhoff problem. We employ a well-known critical point result by Bonanno and Chinn\`ı and show the existence of at least three solutions. We introduce a novel $n$-dimensional vector inequality, Proposition \ref{Prop:2.2}, which is a crucial auxiliary result to obtain the $C^{1}-$regularity of the energy functional.