Suppose $f:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}$ is a Carathéodory function such that for some $a>0$ and $q>1$ we have $|f(x,t)|\leq a(1+|t|^{q-1})$ for all $x\in\mathbb{R}^n$ and $t\in\mathbb{R}$ . Also $\displaystyle\lim_{t\to0}\frac{f(x,t)}{t^{p-1}}=0$ for $p>1$ and all $x\in\mathbb{R}^n$ . Prove that for given $\varepsilon>0$ there exists $C(\varepsilon)>0$ such that$$|F(x,t)|\leq\varepsilon|t|^p+C(\varepsilon)|t|^q$$where $\displaystyle F(x,t)=\int_0^tf(x,s)\,ds$ .
Using the limit near zero one has $\delta>0$ for any $\varepsilon>0$ such that $|f(x,t)|<\varepsilon|t|^{p-1}$ whenever $|t|<\delta$ . Therefore \begin{align*}|F(x,t)|&\leq\varepsilon\int_0^\delta|s|^{p-1}\,ds+a\int_\delta^t(1+|s|^{q-1})\,ds\\&\leq\varepsilon\delta^p+a(|t|+|t|^q-\delta-\delta^q)\end{align*} Rest I am not sure how to determine the $C(\varepsilon)$ in terms of the given constants . Any help is appreciated .