Existence of sign-changing radial solutions with prescribed numbers of zeros for elliptic equations with the critical exponential growth in ℝ²

Abstract

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.