### 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) |
---|---|

Pages (from-to) | 215-228 |

Number of pages | 14 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Rocky Mountain Journal of Mathematics*,

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

**An eigenvalue problem for quasilinear systems.** / Henderson, Johnny; Wang, Haiyan.

Research output: Contribution to journal › Article

*Rocky Mountain Journal of Mathematics*, vol. 37, no. 1, pp. 215-228. https://doi.org/10.1216/rmjm/1181069327

}

TY - JOUR

T1 - An eigenvalue problem for quasilinear systems

AU - Henderson, Johnny

AU - Wang, Haiyan

PY - 2007

Y1 - 2007

N2 - 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 f0 i = lim||u||→00 fi(u)/ψ(||u||), f∞ i = lim ||u||→∞ fi(u)ψ(||u||), i = 1,..., n, f 0 = max{f0 1,..., f0 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.

AB - 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 f0 i = lim||u||→00 fi(u)/ψ(||u||), f∞ i = lim ||u||→∞ fi(u)ψ(||u||), i = 1,..., n, f 0 = max{f0 1,..., f0 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.

UR - http://www.scopus.com/inward/record.url?scp=34249728259&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249728259&partnerID=8YFLogxK

U2 - 10.1216/rmjm/1181069327

DO - 10.1216/rmjm/1181069327

M3 - Article

AN - SCOPUS:34249728259

VL - 37

SP - 215

EP - 228

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

SN - 0035-7596

IS - 1

ER -