American Institute of Mathematical Sciences

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

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.  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

[1]

Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017
[2]

Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363
[3]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[4]

Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747
[5]

Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123
[6]

Lucio Boccardo, Luigi Orsina, Ireneo Peral. A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 513-523. doi: 10.3934/dcds.2006.16.513
[7]

Houda Mokrani. Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1619-1636. doi: 10.3934/cpaa.2009.8.1619
[8]

Farman Mamedov, Sara Monsurrò, Maria Transirico. Potential estimates and applications to elliptic equations. Conference Publications, 2015, 2015 (special) : 793-800. doi: 10.3934/proc.2015.0793
[9]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[10]

Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079
[11]

Luiz F. O. Faria. Existence and uniqueness of positive solutions for singular biharmonic elliptic systems. Conference Publications, 2015, 2015 (special) : 400-408. doi: 10.3934/proc.2015.0400
[12]

Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949
[13]

M. Chuaqui, C. Cortázar, M. Elgueta, J. García-Melián. Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights. Communications on Pure & Applied Analysis, 2004, 3 (4) : 653-662. doi: 10.3934/cpaa.2004.3.653
[14]

Soohyun Bae. Classification of positive solutions of semilinear elliptic equations with Hardy term. Conference Publications, 2013, 2013 (special) : 31-39. doi: 10.3934/proc.2013.2013.31
[15]

Huyuan Chen, Feng Zhou. Isolated singularities for elliptic equations with hardy operator and source nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2945-2964. doi: 10.3934/dcds.2018126
[16]

Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191
[17]

Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943
[18]

Maria Francesca Betta, Olivier Guibé, Anna Mercaldo. Uniqueness for Neumann problems for nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1023-1048. doi: 10.3934/cpaa.2019050
[19]

L. Ke. Boundary behaviors for solutions of singular elliptic equations. Conference Publications, 1998, 1998 (Special) : 388-396. doi: 10.3934/proc.1998.1998.388
[20]

Shuangjie Peng. Remarks on singular critical growth elliptic equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 707-719. doi: 10.3934/dcds.2006.14.707

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences