Let $X$ a reflexive Banach space and $I: X \rightarrow \mathbb{R}$ a functional Gateaux differentiable such that $I \in C^{1}(X,\mathbb{R})$. My first doubt is:
1) If $I$ attains a minimun at the ball $B_{\rho}(0)$ for some $\rho > 0$, that is, exists $u_{0}$ such that $I(u_{0}) = \displaystyle\inf_{B_{\rho}(0)}I$. Is $u_{o}$ a critical point for $I: X \rightarrow \mathbb{R}$?
I know that $I'(u_{0})w = 0$ for all $w \in B_{\rho}(0)$, where $I'(u_{0}):X \rightarrow \mathbb{R}$ is the Gateaux derivative of $I$, then if $u \in X$ , we have
$$I'(u_{0})u= \rho^{-1}||u||I'(u_{0})\dfrac{u}{\rho^{-1}||u||}=0, $$
so $u_{0}$ is a critical point for $I:X \rightarrow \mathbb{R}$. Is my attempt correct?
My other question is:
2) If $I$ attains a minimun in a subset $\mathcal{N} \subset X$, where
$$\mathcal{N} = \lbrace u \in X; I'(u)u=0 \rbrace, $$
is $u_{0}=\displaystyle\inf_{u \in \mathcal{N}}I(u)$ a critical point for $I$?
In this case, $I(u)= \dfrac{1}{2}\displaystyle\int_{\Omega}|\nabla u|^{2} - \displaystyle\int_{\Omega} F(u)$ and $X=H^{1}_{0}(\Omega)$. Moreover, $F(s)=\int_{0}^{s}f(t)dt$, where
$$|f(s)| \leq A|s| + B|s|^{q-1}, 2<q<2^{*}.$$
$2^{*}$ is the Sobolev critical expoent and $A,B>0$.
I'm really thankful for any help.
1) Is not true. You use $I'(u_0) w = 0$ for all $w \in B_\rho(0)$, but this is not true. You only have $I'(u_0) \, (w - u_0) \ge 0$ for all $w \in B_\rho(0)$.
2) Is true under the assumption that $I$ has a global minimizer $\tilde u$ on $X$. In this case, you have $I'(\tilde u) \, w = 0$ for all $w \in X$. Thus, $\tilde u$ belongs to $\mathcal N$, and this implies $\tilde u = u_0$. Hence, $u_0$ is a critical point.