limit of test functions

Question

Hi everyone: Let $ B(x,\frac{1}{n}) $ be a ball of $ \mathbb{R}^{N} $ with $ N\geq2 $. I take a test function $ \phi_{n} $ with compact support in $ B(x,\frac{1}{n}) $, and two sequence of points $ \zeta_{n} $ and $ \eta_{n} $ in $ B(x,\frac{1}{n}) $. Then I let $ n\rightarrow+\infty $. Is it true that $$ \lim_{n\rightarrow+\infty}\frac{\phi_{n}(\zeta_{n})}{\phi_{n}(\eta_{n})}=1? $$

Answer 1

For each $n$ let $\zeta_n$ be a point where $\phi_n$ attains its (without loss of generality positive) maximum and let $\eta_n$ be a point where $\phi(\eta_n) = \frac 12 \phi(\zeta_n)$.

p.s. NO