| Summary: | In this paper, we study the existence of infinitely many
nodal solutions for the following quasilinear elliptic equation
\begin{equation*}
\begin{cases}
-\nabla\cdot\left[\phi'(|\nabla u|^2)\nabla u\right]+|u|^{\alpha-2}u=f(u) , \quad x\in \mathbb{R}^N,\\
u(x)\rightarrow 0, \quad \mbox{as} ~|x|\rightarrow \infty,
\end{cases}
\end{equation*}
where $N\geq2$, $\phi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, $1<p<q<N$, $f\in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R})$ is of subcritical, $q\le\alpha\le p^{*}q'/p'$, let $p^{*}=\frac{Np}{N-p}$, $p'$ and $q'$ be the conjugate exponents respectively of $p$ and $q$. For any given integer $k\geq 0$, we prove that the equation has a pair of radial nodal solution with exactly $k$ nodes.
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