## Abstract

This paper is concerned with a semi-linear elliptic problem with Robin boundary condition: ^{(ε} ∇ ^{∆} w ^{w} · ^{−} ñ + ^{λw} γw ^{1+χ} = ^{=} 0, ^{0} on ^{in Ω} ∂Ω (∗) where Ω ⊂ R^{N}(N ≥ 1) is a bounded domain with smooth boundary, ñ denotes the unit outward normal vector of ∂Ω and γ ∈ R/{0}. ε and λ are positive constants. The problem (∗) is derived from the well-known singular Keller-Segel system. When γ > 0, we show there is only trivial solution w = 0. When γ < 0 and Ω = B_{R}(0) is a ball, we show that problem (∗) has a non-constant solution which converges to zero uniformly as ε tends to zero. The main idea of this paper is to transform the Robin problem (∗) to a nonlocal Dirichelt problem by a Cole-Hopf type transformation and then use the shooting method to obtain the existence of the transformed nonlocal Dirichlet problem. With the results for (∗), we get the existence of non-constant stationary solutions to the original singular Keller-Segel system.

Original language | English |
---|---|

Pages (from-to) | 401-414 |

Number of pages | 14 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 26 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2021 |

## Keywords

- Blowup radius
- Cole-Hopf transformation
- Keller-Segel model
- Nonlocal ODE
- Robin boundary condition

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics