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August  2017, 37(8): 4489-4505. doi: 10.3934/dcds.2017192

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  April 2017

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.
Citation: Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192
References:

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