Criterion for weak convergence to $0$

Question

In the article I have to read it is claimed that if we suppose that we have a (minimizing) sequence (of some functional) in $H^1(\mathbb{R}^n)$ such that $$\iint_{\mathbb{R}^n \times \mathbb{R}^n}u_n^2(x)u_n^2(y)|x-y|^{-1} \rightarrow 0$$ then $u \rightharpoonup 0$ in $H^1(\mathbb{R}^n)$. The functional in question deals with the Choquard-Pekar problem, we find $u$ minimizing $$I_{\lambda} = \text{inf}\int_{\mathbb{R}^n}\frac{1}{2}|\nabla u|^2 + \frac{1}{2}V(x)u^2 - \frac{1}{4}\iint_{\mathbb{R}^n \times \mathbb{R}^n}u_n^2(x)u_n^2(y)|x-y|^{-1}$$ over the set $$K_{\lambda} = \{ u \in H^1(\mathbb{R}^n), ||u||_2^2 = \lambda \}$$ We put some conditions on $I_{\lambda}$ to ensure the problem is well-posed, in particular, this guarantees that $u_n$ (minimizing sequence) bounded in $H^1(\mathbb{R}^n)$. I have no clue on how to see this. Do there exist similar criteria/techniques for the analogue in $W^{1,p}$?