Neumann eigenvalue problems on the exterior domains

Abstract

For p∈(1,∞), we consider the following weighted Neumann eigenvalue problem on B1c, the exterior of the closed unit ball in RN: (0.1)−Δpϕ=λg|ϕ|p−2ϕinB1c,∂ϕ∂ν=0on∂B1,where Δp is the p-Laplace operator and g∈Lloc1(B1c) 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(B1c) into Lp(B1c,|g|) for g in certain weighted Lebesgue spaces. For N>p, we also provide an alternate proof for the embedding of W1,p(B1c) into the Lorentz space Lp∗,p(B1c). Further, we show that the set of all eigenvalues of (0.1) is closed.