Ekeland's variational principle for functional unbounded from below.

Question

It is well-known that if $X$ is a Banach space and $E\in C^1(X)$ is bounded from below, there exists a minimizing sequence $\{u_n\}_{n\geq 1}$ for $E$ in $X$ such that \begin{equation} E(u_n)\rightarrow\inf_{V}E,\quad E'(u_n)\rightarrow 0\text{ in }X'. \end{equation} My question is: Assume now the $C^1$ functional $E$ is not bounded from below, that is $\inf_X E=-\infty$. Consider the open set $X_a=\{u\in X:\|u\|< a\}$, where $a>0$ is a constant. If $E|_{X_a}$ is bounded from below, can we conclude that there exists a minimizing sequence $\{u_n\}_{n\geq 1}\subset X_a$ such that \begin{equation} E(u_n)\rightarrow\inf_{X_a}E,\quad E'(u_n)\rightarrow 0\text{ in }X'. \end{equation} As the first answer shown, this "conjecture" is wrong. But if further we assume that there exists a small positive constant $\rho>0$ such that $\inf_{X_a}E=\inf_{X_{a-\rho}} E$, that is the infimum can not be taken (if it can be taken) near the boundary. Is the existence of the minimizing (PS) sequence right? I'll be grateful for any useful answers!

Answer 1

The answer is no. Consider $X=\mathbb R$ and $f(x)=x$.