  Given $1<p<N$ and two measurable functions $V(r)\geq 0$ and $K(r)>0$, $r>0$,
we define the weighted spaces \[ W=\left\{ u\in
D^{1,p}(\mathbb{R}^N):\int_{\mathbb{R}^N}V\left(\left|x\right|\right)
\left|u\right|^p dx<\infty \right\} , \quad L_{K}^q =L^q(\mathbb{R}^N,K\left(
\left| x\right| \right) dx) \] and study the compact embeddings of the radial
subspace of $W$ into $L_{K}^{q_1}+L_{K}^{q_2}$, and thus into $L_{K}^q$
($=L_{K}^q+L_{K}^q$) as a particular case. Both exponents $q_1,q_2,q$ greater
and lower than $p$ are considered. Our results do not require any compatibility
between how the potentials $V$ and $K$ behave at the origin and at infinity,
and essentially rely on power type estimates of their relative growth, not of
the potentials separately.
