A class of p1(x, .) & p2(x, .)-fractional Kirchhoff-type problem with variable s(x, .)-order and without the Ambrosetti-Rabinowitz condition in Double-struck capital RN

In this article, we study a class of Kirchhoff-type equation driven by the variable s(x, .)-order fractional p(1)(x, .) & p(2)(x, .)-Laplacian. With the help of three different critical point theories, we obtain the existence and multiplicity of solutions in an appropriate space of functions. The main difficulties and innovations are the Kirchhoff functions with double Laplace operators in the whole space Double-struck capital R-N. Moreover, the approach is variational, but we do not impose any Ambrosetti-Rabinowitz condition for the nonlinear term.