Your sentence is weaker than what the limit is saying. As an illustration, $-1/x$ on $x \in (0,\infty)$ has all its values negative, but its limit infimum is zero.
It is generally useful to speak in terms of "tail"s when talking about limits infimum and supremum. Here, a "tail" is: for each choice of $u_0$, restrict the limit as $|u| \rightarrow \infty$ to $|u| > u_0$. Taking the limit on the "complement of a bounded ball" might be more descriptive than "tail", depending on your setting.
For the limit infimum to be negative...
There is a quantity, $m < 0$ such that for each tail of the limit there are infinitely many choices of $(x, u(x))$ in that tail having $F(x, u(x)) < m$.
Note that the limit requires you to sort by magnitude(s) of $u$, but there is no restriction on the choice of $x$ among those $u(x)$ having the same magnitude. So $x$ is part of your choice for points giving the various infima.
We can probably take this a step further...
All tails of the limit have infinitely many $(x,u(x))$ where $F$ is bounded away from and below zero.
"Bounded away from" captures the existence of $m$ in the prior version. It means there is a gap between zero and all the values of $F$ from the called out $(x,u(x))$ points.