Let $M$ be a compact Riemannian $N$-manifold . Let $s\in(0,1)$ and $p\in(1,\infty)$ . Define $(u_n)$ a bounded sequence in $W^{s,p}(M)$ . I want to verify following statements (up to some subsequence) :
$(a)$ $u_n\rightharpoonup u$ weakly as $n\to\infty$ in $W^{s,p}(M)$ .
$(b)$ $\displaystyle\iint_{M\times M}\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))}{(d_g(x,y))^{N+ps}}\rightharpoonup\iint_{M\times M}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{(d_g(x,y))^{N+ps}}$ weakly as $n\to\infty$ in $(L^p(M))'$ .
$(a)$ is evident since $W^{s,p}(M)$ is reflexive . For $(b)$ it is evident that $$U_n:=\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))}{(d_g(x,y))^{\frac{N+ps}{p'}}}\in L^{p'}(M\times M)$$ for $p'=\frac{p}{p-1}$ , hence bounded . Therefore $U_n\rightharpoonup U\in L^{p'}(M\times M)$ weakly. Now I have a problem to show explicitly that $$U=\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{(d_g(x,y))^{\frac{N+ps}{p'}}}$$ Moreover, if we are able to show this, is it enough to conclude the convergence of the functional as in $(b)$ ? Any help is appreciated .