Multiple solutions to Gierer-Meinhardt systems

Abstract

We establish the existence of multiple positive solutions for Gierer-Meinhardt system involving Neumann boundary conditions \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} \Delta u-u+\frac{u^{\alpha _{1}}}{v^{\beta _{1}}} = 0 & \text{in}\;\Omega , \\ \Delta v-v+\frac{u^{\alpha _{2}}}{v^{\beta _{2}}} = 0 & \text{in}\;\Omega , \\ \frac{\partial u}{\partial \eta } = \frac{\partial v}{\partial \eta } = 0 & \text{on}\;\partial \Omega , \end{array} \right. \end{equation*} $\end{document} where $ \Omega \subset \mathbb{R} ^{N} $ ($ N\geq 1 $) is bounded domain having smooth boundary $ \partial \Omega $ and the exponents $ \alpha _{i},\beta _{i}>0 $ $ (i = 1,2) $ satisfy \begin{document}$ \begin{equation*} 0\leq \alpha _{i}-\beta _{i}<\alpha _{i}+\beta _{i}<1. \end{equation*} $\end{document} By means of a new subsupersolution result, shown via Schauder's fixed point theorem and an appropriate truncation, two distinct solutions are obtained. One is zero on the boundary of the domain $ \partial \Omega $ while the second is positive there. At this point, spectral properties of the Laplacian operator have been exploited both in the case of Dirichlet and Neumann boundary conditions. The existence of a third solution is also provided. The approach is is mainly based on Leray-Schauder topological degree. It consists of locating a solution in a perimeter where both previous solutions are not.