I want to check the continuity of the operator $A:X\to X^{*}$ defined by $$\langle A(u),v\rangle=\int_{\Omega}a(x,\nabla u)\cdot\nabla v\,dx-\int_{\Omega}g_k(u)v\,dx\quad\text{ for every }u,v\in X.$$
Here $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ with $N\geq 2$, $X=W_0^{1,p}(\Omega)$ ($p>1$) and $X^{*}$ is the dual space of $X$.
Moreover, $a(x,s)$ is a Caratheodory function defined on $\Omega\times\mathbb{R}^N$ to $\mathbb{R}^N$, and $g_k(s)=\text{min}\{s^{-\delta},k\}$ for $k>0$ and $s>0$ and $g_k(s)=k$ for $s\leq 0$. One can take $A(x,\zeta)=|\zeta|^{p-2}\zeta$ for example.
To proceed, we need to prove that for every $u_n\to u$ in $X$, $$A(u_n)\to A(u)\;\text{ in }X^{*}.$$
I am able to prove that for every $u_n\to u$ in $X$, one has $$\langle A(u_n),v\rangle\to \langle A(u),v\rangle\quad\text{ for every }v\in X.$$ In fact, such a property is called demicontinuity of $A$.
Now from here can I say $A(u_n)\to A(u)$ in $X^{*}$, which will give the continuity of $A$?
Please help me.
Thanks in advance.