Isolated singularity for semilinear elliptic equations
Lei Wei Zhaosheng Feng
Discrete & Continuous Dynamical Systems - A 2015, 35(7): 3239-3252 doi: 10.3934/dcds.2015.35.3239
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.
keywords: eigenvalue minimal positive solution. Hardy potential super-subsolution method
Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system
Rui Peng Xianfa Song Lei Wei
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4489-4505 doi: 10.3934/dcds.2017192
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
Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials
Lei Wei Xiyou Cheng Zhaosheng Feng
Discrete & Continuous Dynamical Systems - A 2016, 36(12): 7169-7189 doi: 10.3934/dcds.2016112
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

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