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

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