Compactness results for the $p$-Laplace equation

Given $1<p<N$ and two measurable functions $V(r)\ge 0$ and $K(r)>0$, $r>0$, we define the weighted spaces $$W=\{u\in D^{1,p}({\mathbb{R}}^N):{\int }_{{\mathbb{R}}^N}V\left(\mid x\mid \right){\mid u\mid }^{p}dx<\mathrm{\infty }\},L_K^q=L^q({\mathbb{R}}^N,K\left(\mid x\mid \right)dx)$$ 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.