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

Pages (from-to) | 1417-1430 |

Number of pages | 14 |

Journal | Mathematische Nachrichten |

Volume | 280 |

Issue number | 12 |

DOIs | |

State | Published - 2007 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Nachrichten*,

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

**On the number of positive solutions of elliptic systems.** / O'Regan, Donal; Wang, Haiyan.

Research output: Contribution to journal › Article

*Mathematische Nachrichten*, vol. 280, no. 12, pp. 1417-1430. https://doi.org/10.1002/mana.200513554

}

TY - JOUR

T1 - On the number of positive solutions of elliptic systems

AU - O'Regan, Donal

AU - Wang, Haiyan

PY - 2007

Y1 - 2007

N2 - The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇u i|p-2∇ui)+λki(|x|) fi(u1, . . . , un) = 0, p > 1, R1 < |x| < R2, ui(x) = 0, on |x| = R1 and R2, i = 1, . . . , n, x ∈ ℝN, where k i and fi, i = 1, . . . , n, are continuous and nonnegative functions. Let u = (u1, . . . , un), φ(t) = |t| p-2t, f0 i = lim∥u∥→0o fi(u)/φ(∥u∥), f∞ i = lim ∥u∥→∞ fi(u)/φ(∥u∥), i = 1, . . . , n, f = (f1 , . . . , fn), f0 = Σi=1 n f0 i and f ∞ = Σi=1 n f∞ i. We prove that either f0 = 0 and f∞ = ∞ (superlinear), or f0 = ∞ 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 f0 = f∞, = 0, or f0 = f∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On The other hand, either f0 and f∞ > 0, or f0 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.

AB - The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇u i|p-2∇ui)+λki(|x|) fi(u1, . . . , un) = 0, p > 1, R1 < |x| < R2, ui(x) = 0, on |x| = R1 and R2, i = 1, . . . , n, x ∈ ℝN, where k i and fi, i = 1, . . . , n, are continuous and nonnegative functions. Let u = (u1, . . . , un), φ(t) = |t| p-2t, f0 i = lim∥u∥→0o fi(u)/φ(∥u∥), f∞ i = lim ∥u∥→∞ fi(u)/φ(∥u∥), i = 1, . . . , n, f = (f1 , . . . , fn), f0 = Σi=1 n f0 i and f ∞ = Σi=1 n f∞ i. We prove that either f0 = 0 and f∞ = ∞ (superlinear), or f0 = ∞ 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 f0 = f∞, = 0, or f0 = f∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On The other hand, either f0 and f∞ > 0, or f0 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.

KW - Annulus

KW - Cone

KW - Elliptic systems

KW - Fixed point theorems

KW - Positive radial solutions

UR - http://www.scopus.com/inward/record.url?scp=34548557540&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548557540&partnerID=8YFLogxK

U2 - 10.1002/mana.200513554

DO - 10.1002/mana.200513554

M3 - Article

AN - SCOPUS:34548557540

VL - 280

SP - 1417

EP - 1430

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

IS - 12

ER -