For p∈(1,∞), we consider the following weighted Neumann eigenvalue problem on B1 c, the exterior of the closed unit ball in RN: −Δpϕ=λg|ϕ|p−2ϕinB1 c, [Formula presented]=0on∂B1,where Δp is the p-Laplace operator and g∈Lloc 1(B1 c)is an indefinite weight function. Depending on the values of p and the dimension N, we take g in certain Lorentz spaces or weighted Lebesgue spaces and show that (0.1)admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue. For this purpose, we establish the compact embeddings of W1,p(B1 c)into Lp(B1 c,|g|)for g in certain weighted Lebesgue spaces. For N>p, we also provide an alternate proof for the embedding of W1,p(B1 c)into the Lorentz space Lp∗,p(B1 c). Further, we show that the set of all eigenvalues of (0.1)is closed. © 2019 Elsevier Ltd