While working on an elliptic problem in $\mathbb{R}^N$, I met an issue that I cannot work out clearly. Assume that we have a continuous function $g \colon \mathbb{R} \to \mathbb{R}$ such that
- $\lim_{s \to 0}\frac{g(s)}{s}=0$;
- $\lim_{s \to +\infty} \frac{g(s)}{s^{2^*-1}}=0$, where $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent in dimension $N\geq 3$.
Now assume I have a sequence $\{u_n\}_n$ of functions that converges strongly to some $u$ in $L^q(\mathbb{R}^N)$, where $q<2^*$. My question is: is it true that $\{g(u_n)u_n\}_n$ converges in $L^1$ to $g(u)u$?
I am in trouble because there is no precise growth condition on $g$, just some asymptotic behavior. It would be standard if $|g(s)| \leq |s|^q$ or something alike.