Sign-changing solutions with prescribed number of nodes for elliptic equations with fast increasing weight

Abstract

In this article, we study the problem
\begin{equation*}
-\Delta u - \frac 12(x.\nabla u)=f(u),
\quad x\in \mathbb{R}^2,
\end{equation*}
where $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is a superlinear continuous function with exponential subcritical or exponential critical growth. The main results obtained in this paper are that for any given integer $k\geq 1$, there exists a pair of sign-changing radial solutions $u_{k}^{+}$ and $u_{k}^{-}$ possess exactly $k$ nodes.