Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in ℝ2

Michinori Ishiwata, Takayoshi Ogawa, Futoshi Takahashi

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the following nonlinearly perturbed version of the elliptic-parabolic system of Keller-Segel type: ∂tu - Δu + ∇ = (u∇v) = 0, t > 0, x ∈ ℝ2, -Δv +v -vp = u, t > 0, x ∈ ℝ2, u(0, x) = u 0(x:) ≥ 0, x ∈ ℝ2, where 1 < p < ∞. It has already been shown that the system admits a positive solution for a small nonnegative initial data in L1(ℝ2) ∩ L2(ℝ2) which corresponds to the local minimum of the associated energy functional to the elliptic part of the system. In this paper, we show that for a radially symmetric nonnegative initial data, there exists another positive solution which corresponds to the critical point of mountain-pass type. The υ-component of the solution bifurcates from the unique positive radially symmetric solution of -Δw + w = wp in ℝ2.

Original languageEnglish
Pages (from-to)1-10
Number of pages10
JournalElectronic Journal of Differential Equations
Volume2009
Publication statusPublished - 2009 Jan 2

Keywords

  • Elliptic-parabolic system
  • Multiple existence

ASJC Scopus subject areas

  • Analysis

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