Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition

Ghergu, Marius

doi:10.1090/S0002-9947-09-04670-4

Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition
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Trans. Amer. Math. Soc. 361 (2009), 3953-3976 Request permission

Abstract:

This paper is concerned with the following Gierer-Meinhardt type systems subject to Dirichlet boundary conditions: \[ \begin {cases} \Delta u - \alpha u + \frac {u^p}{v^q} + \rho (x) = 0,\; u > 0, & \text {in $\Omega $},\\ \Delta v - \beta v + \frac {u^r}{v^s} = 0,\; v > 0, & \text {in $\Omega $}, \\ u=0,\; v=0 & \text {on $\partial \Omega $}, \end {cases} \] where $\Omega \subset \mathbb {R}^N$ ($N\geq 1$) is a smooth bounded domain, $\rho (x)\geq 0$ in $\Omega$ and $\alpha ,\beta \geq 0$. We are mainly interested in the case of different source terms, that is, $(p,q)\neq (r,s)$. Under appropriate conditions on the exponents $p,q,r$ and $s$ we establish various results of existence, regularity and boundary behavior. In the one dimensional case a uniqueness result is also presented.

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