  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.
