  We investigate the existence of infinitely many radially symmetric solutions
for the following nonlinear scalar field equation involving a fractional
$p$-Laplacian, $$(-\Delta_p)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, $$
where $s\in (0,1)$, $2 \leq p < \infty$, $2 \leq N=sp$, $N \in \mathbb{N}$ and
$g\in \mathcal{C} (\mathbb{R}, \mathbb{R})$ is an odd function with both
exponential and polynomial growth. The argument of the proof is based on the
case $N=2 $ of the symmetric mountain pass approach developed by Hirata, Ikoma
and Tanaka in \cite{HIT}.
