Supopse that $\Psi \in C^1(\mathbb{R},\mathbb{R})$, $\Psi(x) \geq - C_0 \, \forall x$ and $|\Psi'(x)| \leq C_1|x|^q+C_2 \, \forall x$. Let $u \in H^1(\Omega)$. Under these assumptions how can I bound $\int_{\Omega}\Psi(u)$ by a constant? ($\Omega \subset \mathbb{R}^n$).
Edit: typo with $r$ instead of an $x$.
Integrating the inequality for $\Psi'$ we obtain $\Psi(x)\le B_1|x|^{q+1}+B_2$ where $B_1,B_2$ are constants. Conversely, the choice $\Psi(x)= |x|^{q+1}$ satisfies the hypotheses. Therefore, the problem could be equivalently stated as: estimate the $L^{q+1}$ norm of $u$.
If $q\le 1$ and $\Omega$ has finite measure, we get $\|u\|_{L^{q+1}(\Omega)}$ bounded by $\|u\|_{L^{2}(\Omega)}$ which is a part of $H^1$ norm.
If $q>1$, we need Sobolev embedding which requires some boundary regularity, such as $\partial \Omega$ being Lipschitz. The embedding places $u$ in $L^p $ with $p=2n/(n-2)$. So, if $1+q\le 2n/(n-2)$, we have an estimate; otherwise there is no bound for $L^{q+1}$ norm. The idea of counterexample was given by Andrew in a comment (you can cap off $|x|^{-\epsilon}$ and smoothen it to get a $C^1$ example with arbitrarily large $L^{q+1}$ norm).