Neumann eigenvalue problems on the exterior domains

For p is an element of(1, infinity), we consider the following weighted Neumann eigenvalue problem on B-1(c), the exterior of the closed unit ball in R-N: -Delta(p)phi = lambda(g)vertical bar phi vertical bar(p-2)phi in B-1(c), partial derivative phi/partial derivative nu = 0 on partial derivative B-1, where Delta(p) is the p-Laplace operator and g is an element of L-loc(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,W-p (B-1(c)) into L-p (B-1(c), vertical bar g vertical bar) for g in certain weighted Lebesgue spaces. For N > p, we also provide an alternate proof for the embedding of W-1,W-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. (C) 2019 Elsevier Ltd. All rights reserved.