How to "explain in words" this expression involving liminf Let $\Omega\subset\mathbb{R}^N$ and $F:\Omega\times\mathbb{R}\to\mathbb{R}$ be a function and consider the expression
$$ \liminf_{| u|\to+\infty} \frac{F(x, u(x))}{|u|^p +| u|^q} <0,$$
for some $p, q >1$. I am trying to "explain in words" this expression in a easy-to-understand way.
Could anyone please suggest me something?
I was thinking something like
"No matter how far you look toward infinity, there will be $|u|$ such that $F(x, u)$ is a negative quantity"
but I am not totally convinced.
Could anyone please help? Thank you in advance!
 A: 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.
