Nodal solutions for a critical Kirchhoff type problem in $\mathbb{R}^N$ - Zhong, Xiao-Jing; Tang, Chun-Lei (Topological Methods in Nonlinear Analysis, 2021)
In the present paper, we concentrate on the following critical Kirchhoff type problem\begin{equation*}-\bigg(a+b\int_{\mathbb{R}^N}| abla u|^{2}dx\bigg)\triangle u+u=|u|^{2^*-2}u+μ|u|^{p-2}u,\quad \text{in } \mathbb{R}^N,\end{equation*}where $N≥ 3$, $a, b> 0$, $p\in (2,\ 2^*)$ and $μ$ is an arbitrary positive parameter. With the help of an equivalent transformation,we first obtain at least one ground state nodal solution with precisely two nodal domains for $N=3$, all $b> 0$ and $N≥4$, $b> 0$ small enough. Moreover, we give a convergence property of ground state nodal solutions as $b\searrow 0$.Besides, we attain infinitely many nodal solutions for $N=3$, $p\in(4, 6)$, all $b> 0$ and $N≥4$, $p\in (2,\ 2^*)$, $b> 0$ sufficiently small, and also establish nonexistence results of nodal solutions for $N≥ 4$ and $b$ large enough.