Nodal solutions for a critical Kirchhoff type problem in $\mathbb{R}^N$

Abstract

In the present paper, we concentrate on the following critical Kirchhoff type problem \begin{equation*} -\bigg(a+b\int_{\mathbb{R}^N}|\nabla u|^{2}dx\bigg)\triangle u+u=|u|^{2^*-2}u+\mu|u|^{p-2}u,\quad \text{in } \mathbb{R}^N, \end{equation*} where $N\geq 3$, $a, b> 0$, $p\in (2,\ 2^*)$ and $\mu$ 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 $N\geq4$, $b> 0$ 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 $N\geq4$, $p\in (2,\ 2^*)$, $b> 0$ sufficiently small, and also establish nonexistence results of nodal solutions for $N\geq 4$ and $b$ large enough.