Let $\Omega$ be an open bounded domain in $\mathbb{R}^N$ and let $J:W_0^{1, 2}(\Omega)\setminus\{0\}\to\mathbb{R}$ be a $\mathcal{C}^1$ functional. Let $(u_n)_n$ be a sequence such that $$ u_n\to u \mbox{ in } L^2(\Omega), \quad u_n\rightharpoonup u \mbox{ in } \ W_0^{1, 2}(\Omega)\quad\mbox{ and }\quad J^{\prime}(u_n)\to 0 \mbox{ in } W^{-1, 2}(\Omega).$$
My question is: under these hypotheses, it is true that $$J^{\prime}(u_n)(u_n -u)\to 0\quad\mbox{ and }\quad J^{\prime}(u)(u_n -u)\to 0?$$
If yes, could anyone please explain me why?
I guess it follows from the assumptions that $J^{\prime}(u_n)\to 0 \mbox{ in } W^{-1, 2}(\Omega)$ and $u_n\rightharpoonup u \mbox{ in } \ W_0^{1, 2}(\Omega)$, but I don't know how to use these informations.
Could anyone please help? Thank you in advance!
If $x_n\to x$ in $X$ and $f_n\rightharpoonup f$ in $X^*$ then $f_n(x_n)\to f(x)$. To prove this, use $$ f_n(x_n)- f(x) = f_n(x_n-x) + (f_n-f)(x) $$