Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\lbrace u_n\rbrace$ tends to $u$ weakly, then $F(u)\leq \liminf F(u_n)$. This is my reasoning: Since $u_n$ weakly converges to $u$ un $H^1$, also converges weakly in $L^2$ and so, (up to a subsequence) one has $u_{\sigma (n)}\rightarrow u$ a.e. Now using this and continuity of $x\mapsto (1-|x|^2)^2$, $f( u_{\sigma (n)})\rightarrow f(u)$ and Fatou's lemma says $$   F(u)=\int f(u)\ dx \leq  \liminf \int f(   u_{\sigma (n)} )\ dx= F(  u_{\sigma (n)})      $$ My question is how can I obtain the same inequality with the original sequence? And, is Rellich theorem useful here? Any answer is welcome! Thank you.