Let $f_n(x)\geq0$ be any sequence of nonnegative $L^1(\mathbb{R}^n)$ functions such that $\int_{\mathbb{R}^{n}}f_n(x)dx=1$ where $dx$ is the Lebesgue measure on $\mathbb{R}^n$. For any $a>1,\epsilon>0$, does there exist a sequence $x_n\in\mathbb{R}^n$ such that $$\lim_{n\to+\infty}a^n\frac{\int_{\|x-x_n\|^2\leq n^{1-\epsilon}}f_n(x)dx}{\int_{\|x-x_n\|^2\leq n}f_n(x)dx}=0$$?

If it does not hold, is there any counter example of the sequence $f_n$? What additional conditions do we need on the sequence $f_n$? A similar question is to show that $$\lim_{n\to+\infty}a^n\frac{\int_{\|x-x_n\|^2\leq n^{1-\epsilon}}\exp(-\|x_n-x\|^2)f_n(x)dx}{\int_{\|x-x_n\|^2\leq n}\exp(-\|x_n-x\|^2)f_n(x)dx}=0$$ ？This can be implied by the first display. Therefore, this is a weaker statement.