On a Class of Non-Local Kirchhoff-Type Double Phase Problem with Variable Exponents and Without the Ambrosetti-Rabinowitz Condition

On a compact Riemannian manifold without boundary, this work deals with a non-local Kirchhoff-type double phase problem with variable exponents and without the Ambrosetti-Rabinowitz. The first objective of this work is to establish the existence of least energy solutions by using the Mountain Pass Theorem and the restriction of Nehari manifold. Furthermore, the second objective is to establish the existence of an infinite number of small and large energy solutions. To obtain these multiplicity results, respectively the Fountain Theorem and the Dual Fountain Theorem are used.