I have a sequence $(u_n)$ such that for a functional $I:W^{1,p}_0(\mathbb{R}^N)\rightarrow \mathbb{R}$ of $C^1-$classe we have $$I'(u_n)u_n=0, \forall n\in \mathbb{N}$$
and $$ \nabla u_n (x) \to \nabla u(x)~ \mbox{and} ~u_n(x) \to u(x) \mbox{a.e in} \mathbb{R}^N. $$
We have also that $I'(u_n)\rightarrow 0$
How to deduce that $$I'(u_n)u\rightarrow 0~\text{and}~ I'(u) u=0$$
Thank you