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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

Lei Wei 1, , Xiyou Cheng 2, and  Zhaosheng Feng 3,

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 $$\lim\limits_{|x-a_i|\rightarrow 0}\frac{w(x)}{|x-a_i|^{-\frac{2}{p-1}}}=\left[\mu+\frac{2}{p-1}\left(\frac{2}{p-1}+2-N\right)\right]^{1/(p-1)}.$$
Keywords: Hardy potential, uniqueness., supersolution, elliptic equations, subsolution, multi-singular points.
Mathematics Subject Classification: Primary: 35J60, 35B40; Secondary: 35B0.
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, 2016, 36 (12) : 7169-7189. doi: 10.3934/dcds.2016112
References:
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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), 731-749. doi: 10.1112/jlms/jdv003.  Google Scholar

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show all references

References:
[1]

D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differential Equations, 224 (2006), 332-372. 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, 227 (2014), vi+85 pp.  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), 317-346. 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), 174-202. 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, World Scientific Publishing, 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), 1-29.  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), 107-124. 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), 637-652. 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), 731-749. 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), 3239-3252. doi: 10.3934/dcds.2015.35.3239.  Google Scholar

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