Ground State Sign-Changing Solutions for Schrödinger-Kirchhoff Equation with Asymptotically Cubic or Supercubic Nonlinearity

Abstract

In this paper, we investigate the existence of a least energy sign-changing solution for the following Schrödinger-Kirchhoff equation

$$\begin{aligned} \qquad \left\{ \begin{array}{l} -(a+b\int _{{\mathbb {R}}^{3}}\vert \nabla u \vert ^2\textrm{d}x)\Delta u+V(x)u=f(x,u), \ x \in {\mathbb {R}}^{3},\\ \\ u \in H^1({\mathbb {R}}^{3}) \end{array} \right. \end{aligned}$$

(SK)

where $a,b>0$ are parameters, $V\in {\mathcal {C}}({\mathbb {R}}^{3}, {\mathbb {R}}^+)$ and $f\in {\mathcal {C}}({\mathbb {R}}^3\times {\mathbb {R}}, {\mathbb {R}})$. The potential function V satisfies some suitable conditions and the nonlinearity f satisfies mild assumptions. By using sign-changing Nehari manifold, we prove that this problem possesses a ground state sign-changing solution with precisely two nodal domains, and its energy is strictly larger than twice that of ground state solutions. Besides, we obtain a convergence property of $u_{b_n}$ as $b_n\searrow 0$. Our results unify asymptotically cubic, which enrich and improve the previous ones in the literature.