Let $E$ be a real Banach space and $I\in\mathcal{C}^1(E;\mathbb{R})$ a functional. Define the Nehari manifold \begin{align} \mathcal{N}=\{u\in E\backslash\{0\}:I'(u)u=0\}, \end{align} where the Frechet derivative of $I$ at $u$, $I'(u)$, is an element of the dual space $E^*$, and we denote $I'(u)$ evaluated at $v\in E$ by $I'(u)v$. Suppose $u\neq 0 $ is a critical point of $I$, i.e., $I'(u)=0$. Then $u\in\mathcal{N}$. Set $S=S_1(0)=\{u\in E:||u||=1\}$ and assume $I(0)=0$. Suppose the following two conditions are true:
- For all $u\in E\backslash \{0\}$ there is a $t_u$ such that if $\phi_u(t)=I(tu)$, then $\phi'_u(t)>0$ for all $0<t<t_u$ and $\phi'_u(t)<0$ for all $t>t_u$,
- There exists a $\delta>0$, independent of $u$, such that $t_u\geq\delta$ for all $u\in E$.
The Nehari manifold has some useful properties as indicated in the references provided here: Question about Nehari manifold. I am in particular interested in justifying that 1 and 2 implies that $\mathcal{N}$ is bounded away from $0$, i.e., for every $u\in\mathcal{N}$ there is a $\rho>0$, independent of $u$, such that $||u||\geq\rho$. I am not sure how to approach the problem.
From 1., there exist for each $v\in S$, a number $s(v)>0$ in such a way that $s(v)v\in\mathcal{N}$ (just solve the equation $\phi'_v(t)=0$ for $t>0$).
Also from 1., $s(v)$ is unique and therefore the application that sends $v\in S$ to $s(v)v\in\mathcal{N}$ is an bijection (in fact, if $\phi''_v(s(v))<0$, it will be a diffeomorphism, while $\mathcal{N}$ is a $C^1$ Banach manifold) with $$||s(v)v||=s(v)||v||=s(v).$$
Therefore $\mathcal{N}=\{s(v)v:\ v\in S\}$ and for each $u\in \mathcal{N}$, we have that (by using 2.) $$||u||=||s(v)v||=s(v)=t_v\ge \delta,$$
where $v=u/||u||$.