JournalspmVol. 65 / No. 1DOI 10.4171/pm/1803

Triple positive solution to the one-dimensional <var>p</var>-Laplacian equation

  • Xuemei Zhang

    North China Electric Power University, Beijing, China
  • Meiqiang Feng

    Beijing Institute of Technology, China
  • Weigao Ge

    Beijing Institute of Technology, China
Triple positive solution to the one-dimensional <var>p</var>-Laplacian equation cover

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Abstract

We obtain sufficient conditions for the existence of at least three positive solutions to the second-order nonlinear delay differential equation with one-dimensional p-Laplacian

(Φp(x'(t)))' + w(t) f(t, x(t), x(t − τ), x'(t)) = 0, t ∈ (0,1), τ > 0,

x(t) = 0, −τ ≤ t ≤ 0,

x(1) = 0,

where Φp(s) is the p-Laplacian operator, i.e., Φp(s) = |s|p−2s, p > 1, (Φp)−1 = Φq, 1/p + 1/q = 1. The arguments are based upon a new fixed point theorem in a cone introduced by Avery and Peterson.