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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.

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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.

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  • $\begingroup$ Which dominating function do you use for DCT? $\endgroup$ – gerw Oct 24 '19 at 18:40
  • $\begingroup$ Thanks for your answer. Im trying now to deal with fatou's lemma. I need to check that $\liminf (1-|u_n|^2)^2 = (1-|u|^2)^2$ $\endgroup$ – Senna Oct 24 '19 at 18:44
  • $\begingroup$ If $H^1$ embeds compactly into L^4 then this works, so $N<4$ $\endgroup$ – daw Oct 24 '19 at 19:09
  • $\begingroup$ Can you explain that, please? $\endgroup$ – Senna Oct 24 '19 at 19:16
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    $\begingroup$ $1$ is not a dominating function. For $u \equiv 3$, you have $(1-u^2)^2 = 64$. For $N \ge 4$, Fatou seems to be the way to go. $\endgroup$ – gerw Oct 25 '19 at 6:02

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