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

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)