I have this lemma+proof and i dont understand why it follows from $J'(u_n)\rightarrow 0$ that $-\Delta_p u_n- f(x,u_n)\rightarrow 0$ such that

$J(u)=\frac1p\int_{\Omega} |\bigtriangledown u|^p dx-\int_{\Omega} F(x,u) dx, u\in W^{1,p}_0(\Omega) $ where $F(x,t)=\int_0^t f(x,s)ds$

This lemma is from the paper

Help me please

Thank you

https://math.stackexchange.com/questions/806766/derivative-of-a-functional