Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system

Rui Peng; Xianfa Song; Lei Wei

doi:10.3934/dcds.2017192

Article Contents

2017, Volume 37, Issue 8: 4489-4505. Doi: 10.3934/dcds.2017192

This issue Previous Article Normalization in Banach scale Lie algebras via mould calculus and applications Next Article Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity

Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system

1.
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
2.
Department of Mathematics, School of Mathematics, Tianjin University, Tianjin 300072, China
3.
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received: November 2016

Revised: March 2017

Published: May 2017

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Abstract

This paper is concerned with the stationary Gierer-Meinhardt system with singularity:

$\left\{\begin{array}{ll} d_1\Delta u-a_1 u+\frac{u^p}{v^q}+\rho_1(x)=0, \ \ & x\in\Omega, \\ d_2\Delta v-a_2 v+\frac{u^r}{v^s}+\rho_2(x)=0,\ \ & x\in\Omega,\\ u(x)>0,\ \ v(x)>0,\ \ & x\in \Omega,\\ \displaystyle u(x)=v(x)=0,\ \ & x\in\partial\Omega, \end{array}\right.$

where $-\infty < p < 1$, $-1 < s$, and $q, r, d_1, d_2$ are positive constants, $a_1, \, a_2$ are nonnegative constants, $\rho_1, \, \rho_2$ are smooth nonnegative functions and $\Omega\subset \mathbb{R}^d\, (d\geq1)$ is a bounded smooth domain. New sufficient conditions, some of which are necessary, on the existence of classical solutions are established. A uniqueness result of solutions in any space dimension is also derived. Previous results are substantially improved; moreover, a much simpler mathematical approach with potential application in other problems is developed.

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Mathematics Subject Classification: Primary: 35J55, 35J75; Secondary: 35Q92.

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