Existence Results for Double Phase Problem in Sobolev–Orlicz Spaces with Variable Exponents in Complete Manifold

In this paper, we study the existence of non-negative non-trivial solutions for a class of double-phase problems where the source term is a Caratheodory function that satisfies the Ambrosetti–Rabinowitz type condition in the framework of Sobolev–Orlicz spaces with variable exponents in complete manifold. Our approach is based on the Nehari manifold and some variational techniques. Furthermore, the Hölder ine-quality, continuous and compact embedding results are proved.