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Certain classes of potentials for p ‐Laplacian to be non‐degenerate
Author(s) -
Zhang Meirong
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410342
Subject(s) - mathematics , degenerate energy levels , exponent , eigenvalues and eigenvectors , laplace operator , dirichlet problem , dirichlet distribution , neumann boundary condition , p laplacian , boundary value problem , dirichlet boundary condition , dirichlet eigenvalue , combinatorics , norm (philosophy) , boundary (topology) , mathematical analysis , pure mathematics , dirichlet's principle , physics , quantum mechanics , philosophy , linguistics , political science , law
Given a positive integer n and an exponent 1 ≤ α ≤ ∞. We will find explicitly the optimal bound r n such that if the L α norm of a potential q ( t ) satisfies ‖ q ‖ L α ( I )< r n then the n th Dirichlet eigenvalue of the onedimensional p ‐Laplacian with the potential q ( t ): (| u ′| p –2 u ′)′ + ( λ + q ( t )) | u | p –2 u = 0 (1 < p < ∞) will be positive. Using these bounds, we will construct, for the Dirichlet, the Neumann, the periodic or the antiperiodic boundary conditions, certain classes of potentials q ( t ) so that the p ‐Laplacian with the potential q ( t ) is non‐degenerate, which means that the above equation with λ = 0 has only the trivial solution verifying the corresponding boundary condition. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)