Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials

We study the quasilinear elliptic equation −div(A(|x|)|∇u|^{p−2}∇u) + V (|x|)|u|{p−2u = K(|x|)f(u) in he whole space. We find existence of nonnegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space X into sum of Lebesgue spaces. Our results do not require any compatibility between how the potentials A, V and K behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of V and K, not of the potentials separately. The nonlinearity f has a double-power behavior.