Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition
Marius Ghergu · Transactions of the American Mathematical Society · 2009
This paper is concerned with the following Gierer-Meinhardt type systems subject to Dirichlet boundary conditions: \[ { Δ u − α u + u p v q + ρ ( x ) = 0 , u > 0 , a m p ; in Ω , Δ v − β v + u r v s = 0 , v > 0 , a m p ; in Ω , u = 0 , v = 0 a m p ; on ∂ Ω , \begin {cases} \Delta u - \alpha u + \frac {u^p}{v^q} + \rho (x) = 0,\; u > 0, & \text {in $\Omega $},\\ \Delta v - \beta v + \frac {u^r}{v^s} = 0,\; v > 0, & \text {in $\Omega $}, \\ u=0,\; v=0 & \text {on $\partial \Omega $}, \end {cases} \] where Ω ⊂ R