Eigenvalues of the Radially Symmetric p ‐Laplacian in R n

Abstract

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