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### Open Access Journals

DCDS

In this paper, we study a class of
semilinear elliptic equations with the Hardy potential. By means of
the super-subsolution method and the comparison principle, we
explore the existence of a minimal positive solution and a maximal
positive solution. Through a scaling technique, we obtain the asymptotic property of positive solutions near the origin.
Finally, the nonexistence of a positive solution is proven when the parameter is larger than a critical value.

DCDS

In this paper, we study the
elliptic equation with a multi-singular inverse square potential
$$-\Delta u=\mu\sum_{i=1}^{k}\frac{u}{|x-a_i|^2}-u^p,\ \ x\in \mathbb{R}^N\backslash\{a_i:i\in K\},$$
where $N\geq 3$, $p>1$ and $\mu>(N-2)^2/4k$.
In our discussions, the domain is the entire space, and the equation contains
multiple singular points.
We not only demonstrate the behavior of positive solutions
near each singular point $a_i$, but also obtain the behavior of positive solutions as $|x|\rightarrow \infty$.
Under suitable conditions,
we show that the equation has a unique positive solution $w$, which satisfies
$$\lim\limits_{|x|\rightarrow\infty}\frac{w(x)}{|x|^{-\frac{2}{p-1}}}=\left[k\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}$$ and
$$\lim\limits_{|x-a_i|\rightarrow 0}\frac{w(x)}{|x-a_i|^{-\frac{2}{p-1}}}=\left[\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}.$$

keywords:
Hardy potential
,
uniqueness.
,
supersolution
,
elliptic equations
,
subsolution
,
multi-singular points

DCDS

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.

keywords:
Gierer-Meinhardt system
,
singularity
,
stationary solution
,
Dirichlet boundary
,
existence
,
uniqueness

## Year of publication

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