We are concerned with the study of the existence and multiplicity of solutions for Dirichlet boundary value problems, involving the (p(m), q(m))− equation and the nonlinearity is superlinear but does not fulfil the Ambrosetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in a complete manifold. The main results are proved using the mountain pass theorem and Fountain theorem with Cerami sequences. Moreover, an example of a (p(m), q(m)) equation that highlights the applicability of our theoretical results is also provided.