I'm studying a paper about elliptc equations. In the paper appears a function $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ such that
$|f(s)| \leq C_{1}|s| + C_{2}|s|^{2^{*} -1}$,
where $C_{1},C_{2} > 0$ and $2^{*}= \frac{2N}{N-2}$ is the Sobolev's critical expoent. (In the case $N=3$)
The authors say that $f$ has subcritical growth. But, my professor say the $f$ has "quasecrítico" (in portuguese, i don't know how say it in english) growth. My professor explains me that are different things, because $f$ has subcritical growth if
$|f(s)| \leq C_{1} + C_{2}|s|^{p-1}$, where $p \in (2,2^{*})$.
So, I'm a bit confused with these definitions.
If someone can help or send some material support, I will appreciate. Thanks.
I am by no means an expert, but to my knowledge this is not a standard nomenclature, rather a descriptive term your professor used to indicate we are including the "critical" case.
By "quasécritico" he likely means quasi-critical in English, which loosely means "partially-critical" or "apparently-critical". This refers to the fact that $p<2^*$ is the subcritical case, $p>2^*$ is, in a sense, the supercritical case, and so the borderline case $p=2^*$ is the critical case. The authors perhaps consider $p\leq2^*$ as the subcritical case?
The reason we consider these cases is usually because of regularity/well-posedness theory, and so this growth bound for $f$ might be called quasi-critical because regularity or well-posedness (of whatever this is in the context of) could depend on a number of things despite this growth bound (that is, different $f$'s with this growth bound might yield different well-posedness results?)