# Another question about weak lower semicontinuity

Let $\Omega$ be a domain bounded, $u_j,u \in H^{1}(\Omega)$ such that $u_j \hookrightarrow u$ in $H^{1}(\Omega)$ (weak) and the functional $$F(u) = \int_{\{u>0\}} f^{+} u dx + \int_{\{u \le 0\}} f^{-}u dx$$ where $f^{+}, f^{-} \in L^{2^{*}}(\Omega)$ with $2^{*} = 2n/(n-2)$. Is true that $F$ is weakly lower simicontinuous? More specifically $$F(u) \le \liminf_{j} F(u_j) ?$$

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 Why post the same question again? math.stackexchange.com/questions/151264/… – user31373 Jun 2 '12 at 17:14