Does the limit of this integral exist?

Question

Let $B(0,R)$ designate the open ball of radius $R>0$ in $\mathbb{R}^{n}$ ($n>1$) centered at the origin. Let also $0<r<R$. We know that by smooth version of Urysohn's lemma there is a test function $\phi$ with compact support in $B(0,R)$ such that $0\leq\phi\leq1$ and $\phi=1$ on $B(0,r)$. Now, can we say that $$\lim_{r\rightarrow R}\int_{B(0,R)\setminus B(0,r)}\Delta\phi(x)dx=0?$$ ($\Delta$ is the laplacian)