New sign-changing solutions to the Hénon problem in dimension $2$

Abstract

In this paper, we construct new sign-changing solutions for the Hénon problem \begin{equation*} \begin{cases} -Δu= |x|^α |u|^{p-1}u,\,\,\,\text{in}\,\,\, Ω, \\[1mm] u=0,\,\,\,\,\,\,\text{on}\,\,\, \partialΩ, \end{cases} \end{equation*} where $\Omega \subseteq {\mathbb{R}}^2$ is a bounded smooth domain containing 0, $\alpha \in \mathbb{R}$ is a positive parameter and $p$ approaches $+\infty$.