Direct method with integral constraint

Question

Let $\Omega\subset\mathbb{R}^n$ be nonempty, open and bounded with $C^1$ boundary. Let $p\in[1,n)$. Let $g\in C(\mathbb{R})$ satisfy $$|g(y)|\leq C(1+|y|^q)$$ for some $C<\infty$ and some $q$ with $1\leq q<p^*$. Let $(u_k)_k$ be a sequence with $u_k\in W^{1,p}(\Omega)$ and $$\int_{\Omega}g(u_k(x))dx=0 \text{ for every } k\in\mathbb{N}.$$ Show: When $u_k$ converges weakly to $u_*$ in $W^{1,p}(\Omega)$, then $$\int_{\Omega}g(u_*(x))dx=0.$$
My attempt:
By using the boundness of weakly convergent sequences, embedding theorems and Arzelà–Ascoli theorem, I obtained a uniformly convergent subsequence $(u_k)_k$ (without renaming), whose limit is $u_*$ due to the uniqueness of the limit. This yields: $$\left\lvert \int_{\Omega}g(u_*(x))dx\right\rvert\leq\int_{\Omega}\lvert g(u_k(x))-g(u_*(x))\rvert dx\leq \varepsilon \lvert\Omega\rvert$$ for $||u_k-u_*||_C<\delta$ using the continuity of $g$. What surprises me now is that I didn't use the growth condition of $g$. Can you assist me with this? Any help is greatly appreciated!
Edit:
Equicontinuity: Morrey yields $W_0^{1,4}(\Omega)\subset C^{0,\frac{1}{4}}(\Omega)$, so $[u_k]_{C^{0,\frac{1}{4}}(\Omega)}\leq c$. This implies $$|u_k(x)-u_k(y)|\leq c|x-y|^{\frac{1}{4}}.$$ Choosing $\delta=(\frac{\varepsilon}{c})^4$: For $|x-y|<\delta$ $$|u_k(x)-u_k(y)|\leq c\left(\left(\frac{\varepsilon}{c}\right)^4\right)^{\frac{1}{4}}=\varepsilon$$

Answer 1

Using the Rellich--Kondrachev theorem, we have \begin{align} \lim_{k \to \infty}\lVert u_k -u_* \rVert_{L^{r}} =0 \, , \end{align} for all $r < p_*$. Furthermore, there exists a subsequence (which we do not relabel) such that $u_k$ converges to $u_*$ a.e. It follows, by continuity of $g$, that $g(u_k)$ converges to $g(u_*)$ a.e. Finally the family of functions $f_k=g(u_k(x))$ are uniformly integrable. By the de la Vallée-Poussin criterion (see https://en.wikipedia.org/wiki/Uniform_integrability#Relevant_theorems), we need to find a superlinear function $G$ such that \begin{align} \sup_{k}\int G(|f_k|)\, \mathrm{d}x < \infty \, . \end{align}
We can choose $G(t)=t^{1+\varepsilon}$ such that $q(1+\varepsilon) <p_*$. We can now apply the Vitali convergence theorem to obtain the desired result (see https://en.wikipedia.org/wiki/Dominated_convergence_theorem Remark 3 or https://en.wikipedia.org/wiki/Vitali_convergence_theorem).