### Abstract

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇u _{i}|^{p-2}∇u_{i})+λk_{i}(|x|) f^{i}(u_{1}, . . . , u_{n}) = 0, p > 1, R_{1} < |x| < R_{2}, u_{i}(x) = 0, on |x| = R_{1} and R_{2}, i = 1, . . . , n, x ∈ ℝ^{N}, where k _{i} and f^{i}, i = 1, . . . , n, are continuous and nonnegative functions. Let u = (u_{1}, . . . , u_{n}), φ(t) = |t| ^{p-2}t, f_{0}^{i} = lim_{∥u∥→0}o f^{i}(u)/φ(∥u∥), f_{∞}^{i} = lim _{∥u∥→∞} f^{i}(u)/φ(∥u∥), i = 1, . . . , n, f = (f^{1} , . . . , f^{n}), f_{0} = Σ_{i=1}^{n} f_{0}^{i} and f _{∞} = Σ_{i=1}^{n} f_{∞} ^{i}. We prove that either f_{0} = 0 and f_{∞} = ∞ (superlinear), or f_{0} = ∞ and f_{∞} = 0 (sublinear), guarantee existence for all λ > 0. In addition, if f ^{i}(u) > 0 for ∥u∥ > 0, i = 1, . . . , n, then either f_{0} = f_{∞}, = 0, or f_{0} = f_{∞} = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On The other hand, either f_{0} and f_{∞} > 0, or f_{0} and f_{∞} < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all die results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point meorems in a cone.

Original language | English (US) |
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Pages (from-to) | 1417-1430 |

Number of pages | 14 |

Journal | Mathematische Nachrichten |

Volume | 280 |

Issue number | 12 |

DOIs | |

State | Published - Sep 18 2007 |

### Keywords

- Annulus
- Cone
- Elliptic systems
- Fixed point theorems
- Positive radial solutions

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Mathematische Nachrichten*,

*280*(12), 1417-1430. https://doi.org/10.1002/mana.200513554