A calculation for an open ball in $\mathbb{R}^N$ and function space.

Question

Let $B_r(x)$ denoting the ball of center $x$ and radius $r>0$. We denote by $\lambda_{1,\,B_\rho(y)}$ the first eigenvalue of $-\Delta$ in $W^{1,\,2}_0\left(B_\rho(y)\right)$ and by $\varphi_{B_\rho(y)}$ the corresponding eigenfunction with the normalization $\int_{B_\rho(y)} \varphi^2_{B_\rho(y)} dx=1$.

I wonder if the following equations are true or not:

• $\varphi_{B_\rho(y)}(x-y)=\rho^{-N/2}\varphi_{B_1(y)}\left((x-y)/\rho\right)$

or

• $\varphi_{B_\rho(y)}(x-y)=\rho^{-N}\varphi_{B_1(y)}\left((x-y)/\rho\right)$

Thank for your reading and helping.