# American Institute of Mathematical Sciences

December  2016, 36(12): 7169-7189. doi: 10.3934/dcds.2016112

## Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials

 1 School of Mathematical and Statistics, Jiangsu Normal University, Xuzhou 221116 2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China 3 School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539

Received  January 2016 Revised  March 2016 Published  October 2016

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 $$\lim\limits_{|x-a_i|\rightarrow 0}\frac{w(x)}{|x-a_i|^{-\frac{2}{p-1}}}=\left[\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}.$$
Citation: Lei Wei, Xiyou Cheng, Zhaosheng Feng. Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7169-7189. doi: 10.3934/dcds.2016112
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##### References:
 [1] D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials,, J. Differential Equations, 224 (2006), 332. doi: 10.1016/j.jde.2005.07.010. Google Scholar [2] F. C. Cîrstea, A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials,, Memoirs of AMS, (2014). Google Scholar [3] F.C. Cîrstea and Y. Du, Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity,, J. Funct. Anal., 250 (2007), 317. doi: 10.1016/j.jfa.2007.05.005. Google Scholar [4] F.C. Cîrstea and Y. Du, Isolated singularities for weighted quasilinear elliptic equations,, J. Funct. Anal., 259 (2010), 174. doi: 10.1016/j.jfa.2010.03.015. Google Scholar [5] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations,, Vol I: Maximum principle and applications, (2006). doi: 10.1142/9789812774446. Google Scholar [6] Y. Du and Z. M. Guo, The degenerate logistic model and a singularly mixed boundary blow-up problem,, Disrete Conin. Dyn. Syst., 14 (2006), 1. Google Scholar [7] Y. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions,, J. London Math. Soc., 64 (2001), 107. doi: 10.1017/S0024610701002289. Google Scholar [8] M. Marcus and L. Véron, Existence and uniqueness results for large solutions of general nonlinear elliptic equations,, J. Evol. Equ., 3 (2003), 637. doi: 10.1007/s00028-003-0122-y. Google Scholar [9] L. Wei and Y. Du, Exact singular behavior of positive solutions to nonlinear elliptic equations with a Hardy potential,, J. Lond. Math. Soc., 91 (2015), 731. doi: 10.1112/jlms/jdv003. Google Scholar [10] L. Wei and Z. Feng, Isolated singularity for semilinear elliptic equations,, Discrete Contin. Dyn. Syst., 35 (2015), 3239. doi: 10.3934/dcds.2015.35.3239. Google Scholar
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