I am not an expert in this kind of thing, so forgive me if I am naive.

I wonder if there is a sequence of functions $ f_t: B _ { mathbb {R} ^ 2} (0, t) to [0,infty]$ (right here $ B (0, t) $ means radius ball $ t $ focused on $ 0) such as, if $ triangle $ is the Laplacian dish

1) $ triangle f_t = 4f_t $ (or more generally $ triangle f_t = cf_t $ for a fixed positive $ c $)

2) $ f_t to 0 $ punctually as $ t to infty $

3) $ f_t (x, y) / f_t (0) to 1 $ punctually as $ t to infty $

In other words, I would like to know if there exists a family of eigenfunctions of the Laplacian, defined on a growing series of marbles, converging towards $ 0 and such as the convergence rate, at least $ infty $, does not depend on the point in question.

Here is a vague idea of why, in my opinion, such a sequence may exist: I can imagine a proper function of the Laplacian on the unit disk whose norm goes up and down slightly from a small amount in small disks . The domain stretch forces the oscillation to go down and finally converge towards $ 0. Maybe I'm looking for a Bessel function.

Thank you.