Weak solvability of nonlinear elliptic equations involving variable exponents.

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 Ambrossetti-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. Note: The second old affiliation of fourth author has been removed. The Acknowledgments' section has been updated. [ABSTRACT FROM AUTHOR]