Eigenvalues of the radially symmetric p-Laplacian in Rn

For the p-Laplacian Delta(p)v=div(\delv\(p-2)delv), p > 1, the eigenvalue problem -Delta(p)v + q(\x\)\v\(p-2)v = lambda\v\(p-2)v in R-n is considered under the assumption of radial symmetry. For a first class of potentials q(r) --> infinity as r --> infinity at a sufficiently fast rate, the existence of a sequence of eigenvalueS lambda(k) --> infinity if k-->infinity is shown with eigenfunctions belonging to L-p (R-n). In the case p = 2, this corresponds to Weyl's limit point theory. For a second class of power-like potentials q(r)-->-infinity as r --> infinity at a sufficiently fast rate, it is shown that, under an additional boundary condition at r = 00, which generalizes the Lagrange bracket, there exists a doubly infinite sequence of eigenvalues lambda(k) with lambda(k) --> +/- infinity if k --> infinity. In this case, every solution of the initial value problem belongs to L-p (R-n). For p = 2, this situation corresponds to Weyl's limit circle theory.