  In this note we complete a previous study, where we got existence results for
the quasilinear elliptic equation \begin{equation*}
  -\Delta w+ V\left( \left| x\right| \right) w - w \left( \Delta w^2 \right)=
K(|x|) g(w) \quad \text{in }\mathbb{R}^{N},
  \end{equation*} with singular or vanishing continuous radial potentials
$V(r)$, $K(r)$. In our previuos study we assumed, for technical reasons, that
$K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper we remove
this obstruction. To face the problem we apply a suitable change of variables
$w=f(u)$ and we find existence of non negative solutions by the application of
variational methods. Our solutions satisfy a weak formulations of the above
equation, but they are in fact classical solutions in $\mathbb{R}^{N} \setminus
\{0\}$. The nonlinearity $g$ has a double-power behavior, whose standard
example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$ ($t>0$), recovering the
usual case of a single-power behavior when $q_1 = q_2$.
