Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$f(u)=\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicomtinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\lbrace u_n\rbrace$ tends to $u$ weakly, then $f(u)\leq \liminf f(u_n)$. I know that the norm $\Vert u \Vert _{H^1}= \Vert u \Vert_{L^2} + \Vert \nabla u \Vert_ {L^2} $ is w.l.s.c., and so $$\int_\Omega |u|^2\ dx$$ is w.l.s.c.
What can be said about $f$? Thanks in advance.
I’m proving that $f$ is weakly continuous. I hope that I’m not wrong.
Assume $u_n \rightarrow u$ weakly. By Reillich, $u_n \rightarrow u$ strongly wrt the $L^2$ norm. So we have a subsequence $u_{p_n}$ that converges ae to $u$.
Then by the dominated convergence theorem, $f(u_{p_n}) \rightarrow f(u)$. Since the same statement can be made for any subsequence of $u_n$ instead, it follows that $f$ is continuous wrt the weak topology.