An eigenvalue problem for quasilinear systems

Johnny Henderson, Haiyan Wang

Research output: Contribution to journalArticle

26 Scopus citations

Abstract

The paper deals with the existence of positive solutions for the n-dimensional 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 denned by Φ(u) = ((μ1),..., (μn)), where u = (u1,...,un), and ψ covers the two important cases ψ(u) = u and ψ(u) = |u|p-2u, p > 1, h(t)=diag [h1(t),...,hn(t)] and f(u) = (f 1(u),...,fn(u)). Assume that fi and h i are nonnegative continuous. For u = (u1,... ,u n), let f0i = lim||u||→00 fi(u)/ψ(||u||), fi = lim ||u||→∞ fi(u)ψ(||u||), i = 1,..., n, f 0 = max{f01,..., f0n} and f = max{f1,... , f n}. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f 0 and f is large enough and the other one is small enough. Our methods employ fixed point theorems in a cone.

Original languageEnglish (US)
Pages (from-to)215-228
Number of pages14
JournalRocky Mountain Journal of Mathematics
Volume37
Issue number1
DOIs
StatePublished - Jun 4 2007

ASJC Scopus subject areas

  • Mathematics(all)

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