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 Schrodinger–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.