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Let $H:=\left\{u\in L^2(R^N):\nabla u \in L^2(R^N)\right\}$ and a functional $$f(u)=\int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2.$$

If $\{u_n\}\subset H$ is a sequence such that $u_n \rightharpoonup u$, is it true that

$$\liminf_{n\rightarrow +\infty}\left[\int_{R^N} |\nabla u_n|^2dx+\left(\int_{R^N} |\nabla u_n|^2dx\right)^2\right]\geq \int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2 \;?$$

2 Answers 2