I have this problem $$ - \operatorname{div}(|\nabla u|^{p-2} \nabla u) =f(u), x\in \Omega $$
With $\Omega $ is bounded in $\mathbb{R}^N$ and $u=0$ on $\partial \Omega $ the condition on $f$ is $f\in W^{-1,p'}(\Omega)$ which is the dual of $ W^{1,p}(\Omega)$.
My problem is to get the variational formulation what is the primitive of $f$?
Do I write$$ I(u)= \frac{1}{p} ||\nabla u||^p -\int F(u) dx$$ where $ F(t)= \displaystyle{\int_0^t f(s) ds}$?
The functional is given by
$$I(u) = \frac1p \|u\|_{W^{1,p}_0}^p - \int_{\Omega} F(u(x)) dx$$
where $F(t) = \int_0^t f(s)ds$.
See this paper by Dinca, Jebelean, and Mawhin. The formula is on page 356 and the preceding discussion is its derivation.