recently i am working in the following question:
Let $F(x,t)$ a Caratheodory function (i.e. $t \mapsto F(x,t)$ is continuous for all $x$ and $x \mapsto F(x,t)$ is Lebesgue mensurable for all $t$ ) such that
$$ |F(x,t)| \leq |t|^{2^\ast},\text{ for all } x\in \mathbb{R}^N, t\in \mathbb{R}, $$ for some $C>0.$
Is it true that the functional
$$ G(u) = \int _{\mathbb{R}^N} F(x,u(x)) dx, u \in D^{1,2} (\mathbb{R}^N), $$
is uniformly continuous in bounded sets of $D^{1,2} (\mathbb{R}^N)$?
If not, what is the assumptions made over $F$ that guarantees the affirmative for the question?
For instance, if $F_t$ exists (the partial derivative in $t$ exists ) and it is Caratheodory, with
$$ |F_t (x,t)| \leq |t|^{2^\ast-1},\text{ for all } x\in \mathbb{R}^N, t\in \mathbb{R}, $$ then, by the Holder's inequality and the Mean Value Theorem, the answer for the question is yes.
In any case, i would appreciate any literature indications.
Thank you for the attention.