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 < ∞} and 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. We consider exponents q1,q2,q that can be greater or smaller than p. 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. We then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear p-Laplace equations of the form - △_p u + V(|x|) |u|^{p-1} u = g(|x|,u) in R^N, 1<p<N, where V and g(|.| ,u) with u fixed may be vanishing or unbounded at zero or at infinity. Both the cases of g super and sub p-linear in u are studied and, in the sub p-linear case, nonlinearities with g(|.| ,0) ≠ 0 are also considered

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