article

Title: A priori estimates of stationary solutions of an activator-inhibitor system

Authors: Huiqiang Jiang and Wei-ming Ni

Issue: Volume 56 (2007), Issue 2, 681-732

Abstract:

$We consider positive solutions of the stationary Gierer-Meinhardt system \begin{eqnarray*} {}&&d_{1}\Delta u-u+\frac{u^{p}}{v^{q}}+\sigma=0\quad\mbox{\ in }\Omega,\\[2pt] {}&&d_{2}\Delta v-v+\frac{u^{r}}{v^{s}}=0hphantom{\ =\ 0}\quad\mbox{in }\Omega,\\[2pt] {}&&\frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=0\hphantom{\ = \ 0 \ =\ 0}\quad\mbox{on }\partial\Omega, \end{eqnarray*} where $\Delta$ is the Laplace operator, $\Omega$ is a bounded smooth domain in $\mathbb{R}^{n}$, $n\geq1$, and $\nu$ is the unit outer normal to $\partial\Omega$. Under suitable conditions on the exponents $p$, $q$, $r$, and $s$, different types of \textit{a priori} estimates are obtained, existence and non-existence results of nontrivial solutions are derived, for both $\sigma>0$ and $\sigma=0$ cases.$