Let $(u_n)_{n\in\mathbb{N}}\subseteq (W^{1,p}_0(\Omega))$, $u_n \rightharpoonup u \in (W^{1,p}_0(\Omega))$ and $B$ be an operator on $(W^{1,p}_0(\Omega))\times(W^{1,p}_0(\Omega))$, where $\Omega \subset \mathbb{R}^d$ is a bounded domain with smooth boundary.
B is defined by $$\langle B(w,u), v\rangle := \int_\Omega ||\nabla u - h(w)||^{p-2}(\nabla u - h(w))\nabla v dx,$$ where $h:\mathbb{R}\to\mathbb{R}^n$ is continuous and bounded and $\limsup_{n \to \infty} \langle B(u_n, u_n), u_n - u \rangle \le 0$ holds.
I want to show that $\langle B(u_n, v), u_n -u\rangle \to 0$ where $v\in (W^{1,p}_0(\Omega))$.
Is it right, that I just have to show that $B(u_n, v)\in(W^{1,p}_0(\Omega))^*$ $\forall n\in\mathbb{N}$?
Then the statement follows from the weak continuity of $(u_n)_{n\in\mathbb{N}}$ (?)