Let $f,f_n: \mathbb{R^3} \to \mathbb{R}$ be positive functions. Suppose $f_n \rightharpoonup f$ in $L^1(\mathbb{R}^3)$ as well as $f_n|v|^k \rightharpoonup f|v|^k$ in $L^1(\mathbb{R}^3)$ for all $0\le k< 2$ (and I know for sure it can not happen for $k=2$). Moreover, the sequence satisfies $$\sup_{n}\int_{\mathbb{R}^3} f_n(v)(1+|v|^2) d\,v < \infty.$$
I have to show that $$\int_{\mathbb{R}^3} f(v) |v|^2 d\,v \le \liminf_n \int_{\mathbb{R}^3} f_n(v)|v|^2 d\,v$$
It should be easy but I can't figure out the exact argument.
It looks like an application of Fatou's lemma, but I don't have pointwise convergence. Maybe it is possible to extract a subsequence such that $f=\liminf_j f_{n_j}$ but I'm not sure.
I know that convexity is sufficient for weak lower semicontinuity when the domain of integration is of finite measure, but here is all $\mathbb{R^3}$. Is it something due to the linearity of the operator?
Thank you very much in advance.