Nonlinear eigenvalue problems for quasilinear systems

J. Henderson, Haiyan Wang

Research output: Contribution to journalArticle

34 Scopus citations

Abstract

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u′))′ + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)φ(p(t)u1), , q(t)φ(p(t)un)), where u = (u1, ..., un), and φ covers the two important cases φ (u) = u and φ (u) = |u| p-2u, p > 1, h(t) = diag[h1(t), ..., hn(t)] and f(u) = (f1(u), ..., fn (u)). Assume that f i and hi are nonnegative continuous. For u = (u 1, , un), let f0i = lim ∥u∥→0 fi(u)/φ(∥u∥),f i = lim ∥u∥→∞ f i(u)/φ(∥u∥), (i=1,...,n), f0 = max{f 01, , f0n} and f = max{f1, , fn}. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f0 and f is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.

Original languageEnglish (US)
Pages (from-to)1941-1949
Number of pages9
JournalComputers and Mathematics with Applications
Volume49
Issue number11-12
DOIs
StatePublished - Jun 1 2005

Keywords

  • Existence
  • Fixed index theorem
  • p-Laplacian

ASJC Scopus subject areas

  • Modeling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

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