## Abstract

The paper deals with the existence and nonexistence of nontrivial nonnegative solutions for the sublinear quasilinear systemdiv (| ∇ u_{i} |^{p - 2} ∇ u_{i}) + λ f_{i} (u_{1}, ..., u_{n}) = 0 in Ω, u_{i} = 0 on ∂ Ω, i = 1, ..., n, where p > 1, Ω is a bounded domain in R^{N} (N ≥ 2) with smooth boundary, and f_{i}, i = 1, ..., n, are continuous, nonnegative functions. Let u = (u_{1}, ..., u_{n}), {norm of matrix} u {norm of matrix} = ∑_{i = 1}^{n} | u_{i} |, we prove that the problem has a nontrivial nonnegative solution for small λ > 0 if one of lim_{{norm of matrix} u {norm of matrix} → 0} frac(f_{i} (u), {norm of matrix} u {norm of matrix}^{p - 1}) is infinity. If, in addition, all lim_{{norm of matrix} u {norm of matrix} → ∞} frac(f_{i} (u), {norm of matrix} u {norm of matrix}^{p - 1}) is zero, we show that the problem has a nontrivial nonnegative solution for all λ > 0. A nonexistence result is also obtained.

Original language | English (US) |
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Pages (from-to) | 186-194 |

Number of pages | 9 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 330 |

Issue number | 1 |

DOIs | |

State | Published - Jun 1 2007 |

## Keywords

- Elliptic system
- Schauder Fixed-Point Theorem
- p-Laplacian

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics