Certain classes of potentials for p -Laplacian to be non-degenerate

✍ Scribed by Meirong Zhang

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 199 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410342

✦ Synopsis

Abstract

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 ‖ < 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)