This paper is concerned with the local minimization problem for a variety of non Frechet-differentiable G\^ateaux functional $J(f)\equiv\int_{\Omega}v(x,u,f)dx$ in the Orlicz-Sobolev space $(W^1_0L_M^*(\Omega),\|.\|_{M})$, where $u$ is the solution of the Dirichlet problem for a linear uniformly elliptic operator with nonhomogenous term $f$ and $\|.\|_{M}$ is the Orlicz norm associated with an N-function~$M$. We use a recent extension of Frechet-differentiability (approach of Taylor mappings see [2]), and we give various assumptions on $v$ to guarantee a critical point is a strict local minimum. Finally, we give an example of a control problem where classical Frechet differentiability cannot be used and their approach of Taylor mappings works.