Compactness and existence results for the p-Laplace equation

Given 10, r>0, we define the weighted spacesW={u∈D1,p(RN):∫RNV(|x|)|u|pdx<∞},LKq=Lq(RN,K(|x|)dx) and study the compact embeddings of the radial subspace of W into LKq1+LKq2, and thus into LKq(=LKq+LKq) 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−△pu+V(|x|)|u|p−2u=g(|x|,u)in RN,1