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 \begin{document}$ ( p( m ), \, q( m ) )- $\end{document} equation and the nonlinearity is superlinear but does not fulfil the Ambrossetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in a complete manifold. The main results are proved using the mountain pass theorem and Fountain theorem with Cerami sequences. Moreover, an example of a \begin{document}$ ( p( m ), \, q( m ) ) $\end{document} equation that highlights the applicability of our theoretical results is also provided.