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

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  • 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.

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


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