### 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) = ((μ_{1}),..., (μ_{n})), where u = (u_{1},...,u_{n}), and ψ covers the two important cases ψ(u) = u and ψ(u) = |u|^{p-2}u, p > 1, h(t)=diag [h_{1}(t),...,h_{n}(t)] and f(u) = (f ^{1}(u),...,f^{n}(u)). Assume that f^{i} and h _{i} are nonnegative continuous. For u = (u_{1},... ,u _{n}), let f_{0}^{i} = lim_{||u||→0}0 f^{i}(u)/ψ(||u||), f_{∞}^{i} = lim _{||u||→∞} f^{i}(u)ψ(||u||), i = 1,..., n, f _{0} = max{f_{0}^{1},..., f_{0}^{n}} and f_{∞} = max{f_{∞}^{1},... , 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 language | English (US) |
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Pages (from-to) | 215-228 |

Number of pages | 14 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - Jun 4 2007 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Rocky Mountain Journal of Mathematics*,

*37*(1), 215-228. https://doi.org/10.1216/rmjm/1181069327