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 ‖equation/tex2gif-inf-2.gif < 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)