### Abstract

We consider the boundary-value problem These assumptions on f imply that the trivial solution (?, u) = (0, 0) is the only solution with ? = 0 or u = 0, and if ? > 0 then any solution u is positive, that is, u > 0 on (0, 1).We prove that the set of nontrivial solutions consists of a C^{1} curve of positive solutions in (0, ?_{max}) × C^{0}[0, 1], with a parametrisation of the form ? ? (?, u(?)), where u is a C^{1} function defined on (0, ?_{max}), and ?_{max} is a suitable weighted eigenvalue of the p-Laplacian (?_{max} may be finite or 1), and u satisfies We also show that for each ? ? (0, ?_{max}) the solution u(?) is globally asymptotically stable, with respect to positive solutions (in a suitable sense). © 2010 Texas State University - San Marcos.

Original language | English |
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Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Electronic Journal of Differential Equations |

Volume | 2010 |

Issue number | 58 |

Publication status | Published - 2010 |

### Keywords

- Nonlinear boundary value problems
- Ordinary differential equations
- P-laplacian
- Positive solutions
- Stability