Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and $p\geq 1$. Suppose that $$ u_n\rightharpoonup u \ \mbox{ in } W_0^{1, p}(\Omega),$$ $$u_n\rightarrow u \ \mbox{ in } L^q(\Omega) \ \mbox{ for all } 1\leq q <p^{\ast},$$ $$u_n\rightarrow u \ \mbox{ a.e. in } \Omega.$$ Could anyone help me to understand why if $$\int_{\Omega} (\vert\nabla u_n\vert^{p-2}\nabla u_n -\vert\nabla u\vert^{p-2}\nabla u)\cdot\nabla (u_n -u) dx\longrightarrow 0\quad \mbox{ as } n\to +\infty,$$ then, as $n\to +\infty,$ $$\Vert u_n - u\Vert_{W_0^{1,p}}\rightarrow 0?$$
I am trying by adding and subtracting some quantities, but it doesn’t work. Maybe the integral above denotes an equivalent norm of $\Vert u_n - u\Vert_{W_0^{1,p}}$?
Could anyone help?
Thank you in advance!