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

Lei Wei; Xiyou  Cheng; Zhaosheng  Feng

doi:10.3934/dcds.2016112

Article Contents

2016, Volume 36, Issue 12: 7169-7189. Doi: 10.3934/dcds.2016112

This issue Previous Article Existence and concentration of solutions for a Kirchhoff type problem with potentials Next Article On large deviations for amenable group actions

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
3.
School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539

Received: December 31, 2015

Revised: February 29, 2016

Published: May 2017

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35J60, 35B40; Secondary: 35B09.

Citation:

Related Papers

Cited by

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.