Consider the problem \begin{cases} -\Delta u = |u|^{p-2}u, \Omega \\ \,\,\,\,\,\,\,\,\,u = 0, \partial \Omega, \end{cases} where $\Omega \subset \mathbb{R}^N$ and $2 < p < 2^{*}$. The functional associated to the problem above is given by $I(u) = \frac{1}{2}||u||_{H^{1}_0(\Omega)}^2 - \frac{1}{p}|u|_{p}^{p}$, that is $$ I'(u)\varphi = \int_{\Omega} \nabla u \cdot \nabla \varphi - \int_{\Omega} |u|^{p-2}u \varphi, \quad \forall u, \varphi \in H^{1}_{0}(\Omega). $$ The Nehari manifold associated to $I$ is defined by $\mathcal{N} = \{u \in H^1_0(\Omega) \backslash\{0\} : I'(u)u = 0\}$. An outhor I'm studying used a version of Ekeland variational principle to minimize the functional $I$ on $\mathcal{N}$. The result is in the paper "On the variational principle, Ekeland, 1974, COROLLARY 3.4:
Let $F$ be a Fréchet-differentiable function and $G$ a $C^{1}$-function such that $$ G(v) = 0 \implies G'(v) \neq 0. $$ Supose moreover that $$ \exists m \in \mathbb{R} : G(v) = 0 \implies F(v) \geq m. $$ Then, for every $\epsilon > 0$, there exists some point $v_{\epsilon}$ and some $\lambda_{\epsilon} \in \mathbb{R}$ such that $G(v_{\epsilon}) = 0$ and $||F'(v_{\epsilon}) - \lambda_{\epsilon} G'(v_{\epsilon})|| \leq \epsilon$.
The author I'm studying applied the agove result with $G(u) = I'(u)u$. However, in this case we have $G(0) = 0$, but $G'(0) = 0$.
What I missed?