  Given two continuous functions $V\left(r \right)\geq 0$ and $K\left(r\right)>
0$ ($r>0$), which may be singular or vanishing at zero as well as at infinity,
we study the quasilinear elliptic equation \[ -\Delta w+ V\left( \left|
x\right| \right) w - w \left( \Delta w^2 \right)= K(|x|) g(w) \quad \text{in
}\mathbb{R}^{N}, \] where $N\geq3$. To study this problem we apply a change of
variables $w=f(u)$, already used by several authors, and find existence results
for nonnegative solutions by the application of variational methods. The main
features of our results are that they do not require any compatibility between
how the potentials $V$ and $K$ behave at the origin and at infinity, and that
they essentially rely on power type estimates of the relative growth of $V$ and
$K$, not of the potentials separately. Our solutions satisfy a weak
formulations of the above equation, but we are able to prove that they are in
fact classical solutions in $\mathbb{R}^{N} \backslash \{ 0\}$. To apply
variational methods, we have to study the compactness of the embedding of a
suitable function space into the sum of Lebesgue spaces
$L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$
($=L_{K}^{q}+L_{K}^{q}$) as a particular case.
  The nonlinearity $g$ has a double-power behavior, whose standard example is
$g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$, recovering the usual case of a
single-power behavior when $q_1 = q_2$.
