Another question about weak lower semicontinuity

Question

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 \begin{equation} F(u) = \int_{\{u>0\}} f^{+} u dx + \int_{\{u \le 0\}} f^{-}u dx \end{equation} where $f^{+}, f^{-} \in L^{2^{*}}(\Omega)$ with $2^{*} = 2n/(n-2)$. Is true that $F$ is weakly lower simicontinuous? More specifically $$ F(u) \le \liminf_{j} F(u_j) ? $$