Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in $R^2$

Abstract

We consider the following nonlinearly perturbed version of the elliptic-parabolic system of Keller-Segel type: $$displaylines{ partial_tu - Delta u+ abla cdot(u abla v)=0,quad t>0,; xinmathbb{R}^2, cr -Delta v+v-v^p=u,quad t>0,; xinmathbb{R}^2,cr u(0,x) =u_0(x)ge 0,quad xinmathbb{R}^2, }$$ where $1<p<infty$. It has already been shown that the system admits a positive solution for a small nonnegative initial data in $L^1(mathbb{R}^2)cap L^2(mathbb{R}^2)$ 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 $v$-component of the solution bifurcates from the unique positive radially symmetric solution of $-Delta w + w = w^p$ in $mathbb{R}^2$.

Keywords

• Multiple existence
• elliptic-parabolic system
• unconditional uniqueness