Existence of Nodal Solutions with Arbitrary Number of Nodes for Kirchhoff Type Equations

Abstract

In this paper, we are interested in the following Kirchhoff type equation

$$\begin{aligned} \left\{ \begin{aligned}&\bigg [a+\lambda \bigg (\int _{{\mathbb {R}}^3}(|\nabla u|^2+V(|x|)u^2)dx\bigg )^{\alpha }\bigg ]\bigg (-\Delta u+V(|x|)u\bigg )=|u|^{p-2}u\quad \text{ in } {\mathbb {R}}^3,\\&u\ \in H^{1}({\mathbb {R}}^3),\\ \end{aligned}\right. \end{aligned}$$(0.1)

where \(a,\lambda >0,\alpha \in (0,2)\) and \(p\in (2\alpha +2,6).\) The potential V(|x|) is radial and bounded below by a positive number. By introducing the Gersgorin Disc’s theorem, we prove that for each positive integer k, Eq. (0.1) has a radial nodal solution \(U_k^{\lambda }\) with exactly k nodes. Moreover, the energy of \(U_k^{\lambda }\) is strictly increasing in k and for any sequence \(\{\lambda _n\}\) with \(\lambda _n\rightarrow 0^+,\) up to a subsequence, \(U_k^{\lambda _n}\) converges to \(U_k^0\) in \(H^{1}({\mathbb {R}}^3)\), which is also a radial nodal solution with exactly k nodes to the classical Schrödinger equation

$$\begin{aligned} \left\{ \begin{aligned}&-a\Delta u+aV(|x|)u=|u|^{p-2}u\quad \text{ in } {\mathbb {R}}^3,\\&u\ \in H^{1}({\mathbb {R}}^3). \end{aligned}\right. \end{aligned}$$

Our results can be viewed as an extension of Kirchhoff equation concerning the existence of nodal solutions with any prescribed numbers of nodes.