  Given two measurable functions $V(r)\geq 0$ and $K(r)> 0$, $r>0$, we define
the weighted spaces \[ H_V^1 = \{u \in D^{1,2}(\mathbb{R}^N):
\int_{\mathbb{R}^N}V(|x|)u^{2}dx < \infty \}, \quad L_K^q =
L^q(\mathbb{R}^N,K(|x|)dx) \] and study the compact embeddings of the radial
subspace of $H_V^1$ 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 super- and sub-quadratic exponents
$q_1$, $q_2$ and $q$ 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. Applications to existence results for
nonlinear elliptic problems like \[ -\triangle u + V(|x|)u = f(|x|,u) \quad
\text{in}\mathbb{R}^N, \quad u \in H_V^1, \] will be given in a forthcoming
paper.
