Weighted p-Laplacian problems on a half-line

Abstract

We study the weighted half-line eigenvalue problem−(|y′(x)|p−1sgny′(x))′=(p−1)(λr(x)−q(x))|y(x)|p−1sgny(x),0≤x<∞, for 1<p<∞, with initial condition y′(0)sin⁡α=y(0)cos⁡α, α∈[0,π), using a modified Prüfer angle ϕ(λ,x). The eigenvalues λk, k≥0, with λk→∞ as k→∞, are characterized by ϕ(λk,x)→(k+1)πp−, and ϕ(λ,x)→(k+1)πp+ if λk<λ<λk+1, as x→∞. We allow the weight r to be locally integrable and definite, semidefinite or indefinite. In the first two cases, the sequence of eigenvalues accumulates at one of ±∞, and in the third, the sequence accumulates at both ±∞. In all cases, solutions y are nonoscillatory on (0,∞) for all λ.