Multiple solutions for a singular fractional Kirchhoff problem with variable exponents

In this work, we studied the multiplicity of solutions for a Kirchhoff problem involving the $\kappa (\xi )$-fractional derivative and critical exponent. More precisely, we transformed the studied problem into an integral equation that lead to the study of the critical point for the energy functional; after that, we presented and proved some properties related to this functional and demonstrated that the energy functional satisfied the geometry of the mountain pass geometry. Finally, by applying the mountain pass theorem for the even functional, we proved that this functional admitted infinitely many critical points, which means that the studied problem has infinitely many solutions.