Let $u$ be a harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 dx \leq C$ for some $C > 0$. Fix a non-negative smooth function $\phi$ which is identically $1$ on $B_1$ and vanishes identically outside $B_2$. Pick $M$ so $|\nabla \phi|, |\Delta \phi| \leq M$. Set $\phi_R(x)=\phi(x/R)$. We have $|\nabla \phi_R(x)|\leq M/R$ and $|\Delta \phi_R(x)|\leq M/R^2$.
Question: I would like to estiamte estimate $$\int_{\mathbb R} \phi_R |\nabla u|^2$$ in a way such that the right hand side goes to zero as $R \to \infty$.
This question is motivated by a post on MathOverflow and this argument would be a step in proving the Liouville theorem. I would like the proof to avoid explicitly using the mean value property.