### Abstract

We consider the diffusive logistic equation with nonlinear boundary conditions δu + λu(1- u) = 0 x ∈Ω, α(u)(∂u/ ∂v) + (1-α(u))u = 0 x ∈ Ω, u 0 for i∈Ω, where Ω is a bounded domain in R
^{N} with v its outer unit normal and A is a positive parameter. The boundary conditions of the equation considered are nonlinear, with the function α satisfying α([0,1]) C [0,1] and increasing. In this work we will study the case α(0) = 0, α′(0) > 0, which implies that, as A varies, the above equation has two continua of solutions, one having Dirichlet boundary conditions, and another one in which each solution is positive at the boundary. We show that the second continuum of solutions may contain infinitely many solutions for a fixed value of λ.

Original language | English (US) |
---|---|

Pages (from-to) | 445-455 |

Number of pages | 11 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - 2011 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Steady state solutions of a logistic equation with nonlinear boundary conditions.** / Cantrell, Robert; Cosner, George; Martínez, Salomé.

Research output: Contribution to journal › Article

*Rocky Mountain Journal of Mathematics*, vol. 41, no. 2, pp. 445-455. https://doi.org/10.1216/RMJ-2011-41-2-445

}

TY - JOUR

T1 - Steady state solutions of a logistic equation with nonlinear boundary conditions

AU - Cantrell, Robert

AU - Cosner, George

AU - Martínez, Salomé

PY - 2011

Y1 - 2011

N2 - We consider the diffusive logistic equation with nonlinear boundary conditions δu + λu(1- u) = 0 x ∈Ω, α(u)(∂u/ ∂v) + (1-α(u))u = 0 x ∈ Ω, u 0 for i∈Ω, where Ω is a bounded domain in R N with v its outer unit normal and A is a positive parameter. The boundary conditions of the equation considered are nonlinear, with the function α satisfying α([0,1]) C [0,1] and increasing. In this work we will study the case α(0) = 0, α′(0) > 0, which implies that, as A varies, the above equation has two continua of solutions, one having Dirichlet boundary conditions, and another one in which each solution is positive at the boundary. We show that the second continuum of solutions may contain infinitely many solutions for a fixed value of λ.

AB - We consider the diffusive logistic equation with nonlinear boundary conditions δu + λu(1- u) = 0 x ∈Ω, α(u)(∂u/ ∂v) + (1-α(u))u = 0 x ∈ Ω, u 0 for i∈Ω, where Ω is a bounded domain in R N with v its outer unit normal and A is a positive parameter. The boundary conditions of the equation considered are nonlinear, with the function α satisfying α([0,1]) C [0,1] and increasing. In this work we will study the case α(0) = 0, α′(0) > 0, which implies that, as A varies, the above equation has two continua of solutions, one having Dirichlet boundary conditions, and another one in which each solution is positive at the boundary. We show that the second continuum of solutions may contain infinitely many solutions for a fixed value of λ.

UR - http://www.scopus.com/inward/record.url?scp=79958726881&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79958726881&partnerID=8YFLogxK

U2 - 10.1216/RMJ-2011-41-2-445

DO - 10.1216/RMJ-2011-41-2-445

M3 - Article

VL - 41

SP - 445

EP - 455

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

SN - 0035-7596

IS - 2

ER -