In this paper, we are concerned with the existence of sign-changing radial solutions with any prescribed numbers of zeros to the following Schrodinger equation with the critical exponential growth:
{
−
Δ
u
+
u
=
λ
u
e
u
2
in
R
2
,
lim
|
x
|
→
∞
u
(
x
)
=
0
,
\begin{equation*} \begin {cases} -\Delta u +u=\lambda ue^{u^2} \quad \quad \text {in } \quad \mathbb {R}^2,\\ \displaystyle \lim _{|x|\to \infty }u(x)=0, \end{cases} \end{equation*}
where
0
>
λ
>
1
0>\lambda >1
. Our proof relies on the shooting method, the Sturm’s comparison theorem and a Liouville type theorem in exterior domain of
R
2
\mathbb {R}^2
. It seems to be the first existence result of sign-changing solution for Schrodinger equation with the critical exponential growth.