Existence of multiple solutions for a nonlinearly perturbed elliptic parabolic system in \(\mathbb{R}^2\)

Ishiwata, Michinori ; Ogawa, Takayoshi ; Takahashi, Futoshi

Electronic journal of differential equations, Tome 2009 (2009)

Résumé

We consider the following nonlinearly perturbed version of the elliptic-parabolic system of Keller-Segel type:

$\displaylines{ \partial_tu - \Delta u+ \nabla \cdot(u \nabla v)=0,\quad t>0,\; x\in\mathbb{R}^2, \cr -\Delta v+v-v^p=u,\quad t>0,\; x\in\mathbb{R}^2,\cr u(0,x) =u_0(x)\ge 0,\quad x\in\mathbb{R}^2, }$

where $1$. 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 $\mathbb{R}^2$.

Détail

Classification : 35K15, 35K55, 35Q60, 78A35
Keywords: multiple existence, elliptic-parabolic system, unconditional uniqueness