On a singularly perturbed semi-linear problem with robin boundary conditions

Qianqian Hou, Tai Chia Lin, Zhi An Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)

Abstract

This paper is concerned with a semi-linear elliptic problem with Robin boundary condition: w w · ñ + λw γw 1+χ = = 0, 0 on in Ω ∂Ω (∗) where Ω ⊂ RN(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 = BR(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 languageEnglish
Pages (from-to)401-414
Number of pages14
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume26
Issue number1
DOIs
Publication statusPublished - 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

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