Nodal solutions for quasilinear equations of Kirchhoff type in \({\mathbb {R}}^3\)

In this paper, we consider a kind of quasilinear equation of Kirchhoff type in \({\mathbb {R}}^3\) ${\mathbb{R}}^3$ \(\begin{aligned} \left\{ \begin{aligned}&a(u)\Delta u+\frac{1}{2}a'(u)|\nabla u|^2-V(x)u+\int _{{\mathbb {R}}^3}b(u)|\nabla u|^2\text{ d }x(b(u)\Delta u\\&+\frac{1}{2}b'(u)|\nabla u|^2)+f(u)=0,\, \text{ in }\, \, {\mathbb {R}}^3,\\&u(x)\rightarrow 0, \quad \text{ as } \,\,\ \ |x|\rightarrow \infty , \end{aligned} \right. \ \ (P) \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& a(u)\mathrm{\Delta }u+\frac{1}{2}{a}^{\prime }{(u)|\mathrm{\nabla }u|}^2-V(x)u+{\int }_{{\mathbb{R}}^3}{b(u)|\mathrm{\nabla }u|}^2\text{d}x(b(u)\mathrm{\Delta }u\hfill \\ \hfill & +\frac{1}{2}{b}^{\prime }{(u)|\mathrm{\nabla }u|}^2)+f(u)=0,\text{in}{\mathbb{R}}^3,\hfill \\ \hfill & u(x)\to 0,\text{as}|x|\to \infty ,\hfill \end{array}\right)(P)\end{array}$$ where the potential function V is a radial function, the coefficients a and b are of quadratic growth, and the nonlinear term f may be either of subcritical growth or of critical growth. By using the Nehari method we prove that for any given positive integer k the problem has two radial solutions, each with k nodal domains exactly.