  Given $N\geq 3$, $1<p<N$, two measurable functions $V\left(r \right)\geq 0$,
$K\left(r\right)> 0$ and a continuous function $A(r) >0$ ($r>0$), we study the
quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2}
\nabla u\right) u+V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad
\text{in }\mathbb{R}^{N}. \] We find existence of nonegative solutions by the
application of variational methods, for which we have to study the compactness
of the embedding of a suitable function space $X$ 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. Our results do not require any
compatibility between how the potentials $A$, $V$ and $K$ behave at the origin
and at infinity, and essentially rely on power type estimates of the relative
growth of $V$ and $K$, not of the potentials separately. The nonlinearity $f$
has a double-power behavior, whose standard example is $f(t) = \min \{ t^{q_1
-1}, t^{q_2 -1} \}$, recovering the usual case of a single-power behavior when
$q_1 = q_2$.
