Compactness and existence results in weighted Sobolev spaces of radial functions. Part I: compactness

Given two measurable functions and , , we define the weighted spaces and study the compact embeddings of the radial subspace of into , and thus into () as a particular case. Both super- and sub-quadratic exponents , and are considered. Our results do not require any compatibility between how the potentials and 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 in , , will be given in a forthcoming paper.