Neumann eigenvalue problems on the exterior domains. (October 2019) - Anoop, T.V.; Biswas, Nirjan
Abstract: For p ∈ ( 1, ∞ ), we consider the following weighted Neumann eigenvalue problem on B 1 c, the exterior of the closed unit ball in R N : (0.1) − Δ p ϕ = λ g | ϕ | p − 2 ϕ in B 1 c, ∂ ϕ ∂ ν = 0 on ∂ B 1, where Δ p is the p -Laplace operator and g ∈ L l o c 1 ( B 1 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 W 1, p ( B 1 c ) into L p ( B 1 c, | g | ) for g in certain weighted Lebesgue spaces. For N > p, we also provide an alternate proof for the embedding of W 1, p ( B 1 c ) into the Lorentz space L p ∗, p ( B 1 c ) . Further, we show that the set of all eigenvalues of(0.1) is closed.