This paper is concerned with the local minimization problem for a variety of non Frechet-differentiable G\^ateaux functional $J(f)\equiv \int_{Q}v(x,u,f)\,dx$ in the Sobolev space $(W^{1,2}_0(Q),\|\cdot\|_p)$, where $u$ is the solution of the Dirichlet problem for a linear uniformly elliptic operator with nonhomogenous term $f$ and $\|\cdot\|_{p}$ is the norm generated by the metric space $L^p(Q)$, $(p>1)$. We use a recent extension of Frechet-differentiability (approach of Taylor mappings, see [5]), and we give various assumptions on $v$ to guarantee a critical point to be 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.