Compactness results for the p-Laplace equation

Given 1<p<N and two measurable functions V(r)≥0 and K(r)>0, r>0, we define the weighted spaces W = {u ∈ D^{1,p}(R^N) : ∫_R^N V(|x|)|u|^p dx < ∞}, L_K^q = L^q(R^N, K(|x|)dx) and study the compact embeddings of the radial subspace of W into L_K^q1+L_K^q2, and thus into L_K^q (=L_K^q+L_K^q) as a particular case. Both exponents q1,q2,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.