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Eigenvalues of the Radially Symmetric p ‐Laplacian in R n
Author(s) -
Brown B. M.,
Reichel W.
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070300512x
Subject(s) - eigenvalues and eigenvectors , eigenfunction , laplace operator , combinatorics , mathematics , sequence (biology) , limit (mathematics) , p laplacian , limit point , mathematical analysis , physics , boundary (topology) , symmetry (geometry) , mathematical physics , boundary value problem , quantum mechanics , geometry , biology , genetics
For the p ‐Laplacian Δ p υ = div:(| ∇υ| p −2 ∇υ), p >1, the eigenvalue problem −Δ p υ + q (| x |)|υ| p −2 υ = λ|υ| p −2 υ in R n 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 λ k →∞ if k →∞ 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 )→−∞ 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 λ k with λ k → ±∞ if k →±∞. 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.