Existence of sign-changing radial solutions with prescribed numbers of zeros for elliptic equations with the critical exponential growth in \mathbb{R}^2.

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: \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<\lambda <1. Our proof relies on the shooting method, the Sturm's comparison theorem and a Liouville type theorem in exterior domain of \mathbb {R}^2. It seems to be the first existence result of sign-changing solution for Schrodinger equation with the critical exponential growth. [ABSTRACT FROM AUTHOR]