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

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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

Abstract
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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 and Continuous Dynamical Systems, 2016, 36 (12) : 7169-7189. doi: 10.3934/dcds.2016112

References:

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[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.
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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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