For a bounded region \(\Omega\subset\mathbb R^N\) and a function \(p\in C(\bar\Omega)\) such that \(p(x)\geq 2\) in \(\Omega\), the author studies the Dirichlet problem \[ -\text{div}(| \nabla u(x)| ^{p(x)-2}\nabla u(x))=F_{u}(x,u(x)),\quad u| _{\partial\Omega}=0 \] looking for weak solutions belonging to the generalized Orlicz-Sobolev Space \(W_{0}^{1,p(x)}(\Omega)\). Under suitable convexity assumptions on \(F\) he obtains the existence of a solution by applying a dual variational method.