If $u_m\rightarrow u$ in $H^1(\mathbb{R}^n)\times H^1(\mathbb{R}^n)$(weak) and $u_m\rightarrow u$ in $L^4(\mathbb{R}^n)\times L^4(\mathbb{R}^n)$, denote $u_m=(u_{m,1},u_{m,2})$ and $u=(u_{1},u_{2})$, then whether could we get:
$\liminf\int_{\mathbb{R}^n} u_{m,1}^2u_{m,2}^2\,dx \rightarrow \int_{\mathbb{R}^n} u_1^2u_2^2\,dx$ under the condition that
$\int_{\mathbb{R}^n} |\nabla u_{m,1}|^2+u_{m,1}^2\,dx -\int_{\mathbb{R}^n} u_{m,1}^4+u_{m,1}^2u_{m,2}^2\,dx$ and
$\int_{\mathbb{R}^n} |\nabla u_{m,2}|^2+u_{m,2}^2\,dx -\int_{\mathbb{R}^n} u_{m,1}^4+u_{m,1}^2u_{m,2}^2\,dx$
I encountered this conclusion in a paper but I can't figure out how to prove it. Thanks for any help!!