From Prove Liouville theorem without using mean value property the following question arises: To prove the Liouville theorem

If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 dx \leq C$ for some $C > 0$, then $u$ is constant.

is it possible to use the following reasoning?

Fix a non-negative smooth function $\phi$ which is identically $1$ on $B_1$ and vanishes identically outside $B_2$. Pick $M$ so $|\Delta \phi| \leq M$. Set $\phi_R(x)=\phi(x/R)$. We have $|\nabla \phi_R(x)|\leq M/R$ and $|\Delta \phi_R(x)|\leq M/R^2$, then estimate $$\int_{\mathbb R} \phi_R |\nabla u|^2$$ in a way such that the right hand side goes to zero as $R \to \infty$.

Unfortunately, I'm not able to complete this proof, but it seems that it should work.