Ground state and nodal solutions for critical Schrödinger–Kirchhoff-type Laplacian problems

Abstract

In this paper, we are interested in the existence of ground state nodal solutions for the following Schrödinger–Kirchhoff-type Laplacian problems:

$$\begin{aligned} -M\bigg (\int _{{\mathbb {R}}^{3}}|\nabla u|^{2}\mathrm{{d}}x\bigg )\Delta u+V(x)u=|u|^{4}u+ k f(u),\;x\in {\mathbb {R}}^{3}, \end{aligned}$$

where $M(t)=a+bt^\gamma$ with $0<\gamma <2$, $a,b>0$ and the nonlinear function $f\in C({\mathbb {R}},{\mathbb {R}})$. By the nodal Nehari manifold method, for each $b>0$, we obtain a least energy nodal solution $u_{b}$ and a ground state solution $v_b$ of this problems when $k\gg 1$. Our results improve and extend the known results of the usual case $\gamma =1$ in the sense that a more wider range of $\gamma$ is covered.